The spatial proximity between the origin and destination significantly influences the utilization of supporting facilities (Dijk and Wulp, 2010; Lira and March, 2021). It serves as a pivotal factor for evaluating the rationalization of transportation systems, land usage, and the arrangement of each supporting facility within the industrial parks (Gehl, 2002). Notably, the equivalent distance coefficient finds direct application in converting paths between mountainous and plain terrains. In studies concerning facility layout in mountainous regions, the extent of user fatigue while walking often acts as a criterion for assessing “spatial proximity” (Minetti et al., 2002). To establish a quantitative approach for determining the service radius length while considering terrain undulation, the equivalent distance function is introduced:

k i is the equivalent distance coefficient, which is defined as the ratio of the horizontal section distance to the slope of the inclined section when reaching the same degree of walking fatigue under the influence of mountainous areas, expressed as follows:

k i represents the equivalent distance coefficient. l i represents the distance of the horizontal section when the same degree of walking fatigue is reached. l i ' is the horizontal distance of the inclined section when the same degree of walking fatigue is reached.

As defined above, in order to further derive the equivalent distance coefficient, the concept of oxygen consumption is introduced. Oxygen consumption often serves as a measure to assess fatigue levels (American College of Sports Medicine, 2006). Thus, in an optimal scenario, oxygen consumption becomes a tool for evaluating “spatial proximity”, in other words, the strategic arrangement of supporting facilities. Noteworthy scholars such as Minetti and Steffen have proposed a relationship between road sections and energy consumption during comfortable walking speeds, particularly when the slope ranges from −45% to 45% (Sitte, 1990; Minetti et al., 2002; Held et al., 2023):

v 0 ' represents the oxygen consumption per unit weight distance L/(kg·m), and G represents the slope.

The relationship about the equivalent coefficient can be deduced from Formula 1, with the oxygen consumption per unit weight distance on the horizontal section v 0 ' (0) = 0.08 L/(kg·m). The calculation of the absolute value of the slope can be deduced from the calculation formula for k i v , as follows:

k i v represents the equivalent distance coefficient based on oxygen consumption; G represents the road slope. The quantitative research formula based on the energy consumption relationship is more complex, and the calculation results under ideal conditions. Therefore, it is impossible to study the layout of supporting facilities in mountainous industrial parks only based on the fatigue of oxygen consumption (Dijk and Wulp, 2010).

In industrial parks situated on flat terrains, the service radius of supporting facilities is typically determined by multiplying the upper limit of suitable walking distance by the detour coefficient, PDR (Stangl, 2012). Environmental issues often find resolution through experimental methodologies (Clobert et al., 2018). The detour coefficient pertains to the ratio between the actual distance and the straight-line distance due to the twists and turns in the path caused by the park’s ground texture.

The formula is as follows:

r represents the service radius taken, l represents the limiting walking distance, and PDR represents the detour coefficient.

A simple circular area is insufficient for accurately representing the appropriate service range of supporting facilities within mountainous parks (Tao et al., 2017). Due to the diverse slopes in different directions leading to varying effects on walking fatigue, we aimed to systematically develop a quantitative approach for strategically placing supporting facilities in mountainous terrain. In this regard, we presented a visual analysis method of the rational arrangement of these supporting facilities within mountainous industrial parks. The process of analysis is depicted in Figure 1 .

The walking path of the mountainous industrial parks is also influenced by the detour coefficient; with reference to Formula 6, the appropriate one-way horizontal walking distance threshold is as follows:

S represents the appropriate one-way horizontal walking distance threshold.

Assuming the planar location B of the supporting facilities, we drew the radial line R m with B as the center of the circle in the plane to a certain angle, and each segment was named D ₁ to D_n .

We projected the starting and ending points of D_n onto the terrain surface and connected the two projection points to obtain the line segment D n ' and the height difference Δ H _ n between the two points. We can obtain the slope in the vertical direction D n ' , which is the average slope of the section:

G_n represents the slope of the road section D n ' , and Δ H _ n represents the absolute value of the height difference between the two ends of D_n.

Taking G_n into Formula 8, the equivalent distance coefficients of road sections D_n can be obtained, and thus the equivalent walking distance of each road section in the radial direction

where l n ' stands for the road section D_n equivalent walking distance.

By accumulating l n ' , the equivalent walking distance L m ' can be obtained in the direction of radial R_m starting from point B.

The horizontal distance is represented on the X-axis, and the equivalent walking distance is indicated on the Y-axis. We establish the optimal walking distance threshold as S. When Y equals S, the corresponding horizontal distance aligns with the furthest point Pm, which is considered feasible for walking within this radial distribution. By connecting the acquired Pm points from each radiation direction, we can define a suitable service range originating from point B. Figure 1 illustrates the strategic arrangement of supporting facilities within the mountainous industrial park.

In mountainous regions, the presence of large and extended steep slopes and steps complicates the design of supporting facility layouts in parks (Jin et al., 2013). Pedestrian bridges and underpasses in these areas can span several times the walking distance of flat pathways (Zhai and Zhou, 2019). By examining the equivalent distance of ascents and descents along mountainous roads, we have developed a visualization program utilizing the programming languages Rhino and Grasshopper to logically arrange supporting facilities within the mountainous park. This enables the practical application of our research findings to the layout design of these park facilities. Rhino 7.0 is capable of creating and editing terrain surfaces, while Grasshopper 1.0.0007 facilitates modular visual programming.

The rational layout of supporting facilities in mountainous industrial parks, viewed from a user’s perspective, necessitates consideration of factors such as functionality, service objects, usage frequency, and spatial proximity (Ye et al., 2020; Han et al., 2021). In accordance with the “Urban Residential Area Planning and Design Standards”, amenities such as retail, dining, and leisure facilities should be placed within the 5-minute life circle’s service radius. Here, L0 signifies the distance achievable in a 5-minute walk, depicting the actual path distance between two points without accounting for path slope (Ren et al., 2020), while the service radius assumes the straight-line distance between points, denoted as R0 when the path slope is omitted. A visualization analysis of the mountainous park site was undertaken to regulate the theoretical PRD values within the range of 1.1 to 1.5 (Shen et al., 2020).

The program interface is categorized into three sections: input, calculation, and output. The interface of the visual analysis program for the rational facility layout is depicted in Figure 2 . Within the input section, adjustments can be made to the terrain surface, the planar positions of the supporting facilities can be assumed, and the range radius, standard service radius, and ray segmentation modulus can be measured. Meanwhile, the output section allows for color adjustments of the reasonable service range surface area, along with displaying the area and coverage of the range. The calculation segment comprises standard analysis units, with each unit corresponding to a radiation line.

Chongqing, situated in southwestern China (see Figure 3 ), boasts an urbanization rate of 65.5%, rendering it a renowned mountainous city within the country. Its terrain is predominantly hilly and mountainous, constituting 76% of the total land area. Significantly, Chongqing holds a pivotal role in the Western Development strategy and serves as a crucial nexus between the “Belt and Road” initiative and the Yangtze River Economic Belt. The region’s ecology, as noted by Su et al. (2022), is intricate. Liangjiang New Area, encompassing 1,200 km², falls under the jurisdiction of Chongqing City as a sub-provincial and national-level development zone. Notably, it stands as China’s first national-level new area in the inland region. Dadi Industrial Park, an integral component of this area, is situated within the Shuitu Cluster of Chongqing Liangjiang New Area. Dadi Industrial Park is an integral component of the Chongqing Liangjiang New Area’s science and technology innovation base, a key project initiated by the Ministry of Science and Technology. As a prominent biopharmaceutical-focused industrial park in Chongqing, it has successfully attracted investments from over 150 enterprises. It serves as a mountainous, medium-sized biomedical enterprise, with an expansive plot spanning 102,474 m². This industrial park finds itself proximate to the city’s main thoroughfare to the west, urban bypasses and residential communities to the north, and additional industrial land and urban bypasses to the east. The concave flood discharge canal defines its southern border, while a city park stands opposite the flood drainage canal in the southwest. Following topographical adjustments, the park is partitioned into three distinct platforms: one on the west, another on the south, and a central platform. Notably, the west and south platforms lie in close proximity to the primary city road and the flood discharge channel, respectively. These areas exhibit an elevation of approximately 4.5 m lower than that of the central platform. The site layout can be visualized in Figure 3 .

We conducted walking experiments across diverse slope sections to quantify walkers’ perceived fatigue and derived the equivalency distance coefficient formula through data analysis. The experiment involved 10 adults, with an equal gender distribution (five men and five women), aged between 25 and 65 years. All participants were in good health without any medical history and represented the primary users of the industrial park. Five sections with different slopes were selected: Section A (0.5%), Path B (4.6%), Path C (8.3%), Path D (15.2%), and Path E (24.8%). We chose a suitable weather and time period and asked walkers to walk to and fro at their normal speed and manner. Heart rate measurements and the Brog Scale were used as auxiliary judgments to enhance the accuracy of walkers’ perception of their walking fatigue.

In accordance with pertinent regulations, the Chongqing Dadi Industrial Park parcel allows for ancillary structures covering up to 7% of the area. These structures encompass office, recreational, and dining facilities. The layout of these supporting facilities is depicted in Figure 5 , illustrating their designated positions within the park. During program deliberations, four potential locations for these supporting facilities were established, considering the urban context. Position 1 situates itself at the primary entrance square on the western terrace. Position 2 is found on the southwest terrace fortress, approximately 10 m lower in elevation compared to the western terrace, overlooking the municipal garden landscape to the south. Position 3 is centered within the park’s horizontal and vertical axes, positioned at the square within the central terrace. Finally, Position 4 is situated at the midpoint of the southern terrace, representing the park’s southernmost terrace. For clarity, the location of these supporting facilities is depicted in Figure 5 .