Reducing Labor Intensity when Computing Optimal Technical Characteristics of Aerial Ropeways

The article develops a procedure for optimizing the technical characteristics of ropeways – the step and the height of intermediate towers, and carrying ropes tension force. The optimization problem was based on the minimization of the tower structures cost. The reduction of computing labor intensity is based on the fact that the position of the minimum point of the objective function will be tied to one of the optimization restrictions. This allowed us to propose two ways to reduce the labor intensity of computing: a) reduction in the dimension of the optimization problem; b) replacement of the search for the minimum of the objective function with the solution of the nonlinear algebraic equation. The article shows that the proposed algorithm has increased computational efficiency. The algorithm allows us to obtain the same optimal values of technical characteristics of ropeways as in the solution of the previously developed optimization task but using simpler mathematical methods.


INTRODUCTION
Ropeways as overland transport systems have been widely used in many countries of the world as a continuous mode of transportation for freight and passenger travel [1][2][3]. Freight ropeways are used in many sectors of economy (mining, coal mining, chemicals production, metallurgy, energy, forestry, and agriculture) with the purpose of conveying process equipment within individual enterprises or providing logistic links to other enterprises, transport and storage terminals or construction sites [4,5].
Passenger ropeways were originally used to improve the comfort of travel for people within sports or recreational facilities [2,6]. However, since the 1990s, passenger ropeways have become quite actively used as extra-street mode of public transportation in large metropolitan cities [7,8], and also as passengers traveling facility within tourist, recreational and natural territories [9,10]. Currently, passenger ropeways are being operated in every continent, but most of them are located in Europe (Germany, Austria, France, Italy, Switzerland, Russia), America (Bolivia, Colombia, Venezuela) and Asia (China, India, Vietnam). A detailed overview of ropeways use in different countries can be found in [2,11,12].
The use of passenger aerial ropeways as urban transport can solve transport problems which are impossible to be fixed by traditional land transport (wheeled, rail or conveyor ones) [13,14]. The advantages of ropeway transport are particularly evident when such factors as terrain, high density of residential or industrial buildings and various urban planning restrictions hinder the development of land transportation [2,15]. Also, according to existing feasibility studies [7,13,16] ropeway transport is more economically and environmentally friendly than land transport. Therefore, it can be said that passenger aerial ropeways in an urban environment are a socially attractive and economically efficient type of transport. As shown in [3,11,17], passenger ropeways are now increasingly being regarded as an efficient alternative to traditional land public transport modes in large cities.

ANALYSIS OF THE RESEARCH PROBLEM
The problem of the study of passenger aerial ropeways is complex, as it has several scientific aspects -technical, economic, social, and legal ones. Most of the existing studies are devoted to engineering problems of design and calculation of basic structural elements of ropeways, for example, analysis of dynamics and strength of carrying ropes [18,19] or risk analysis of ropeways operation [20]. Also, for example, [15] considered the issues of social and economic impact of ropeways construction on the development of adjacent territories, [21] -issues of legal registration of the ownership for air space and land for ropeways.
Economic challenges of ropeways construction have not yet been widely considered, although they determine the prospects of urban transport infrastructure modernization based on ropeways. Economic studies tend to assess the economic impact of replacing the existing land transport system with an alternative ropeway. The issues of identifying most appropriate technical characteristics of ropeways based on the need to minimize construction costs, have not yet been studied. Such topics are discussed, for example, in [22][23][24].
Construction of passenger ropeways in a highly urbanized environment of a large city or metropolis is a very expensive technical and economic task [2,25]. The cost of construction includes the cost of survey, construction, installation and design works, purchase of mechanical equipment, development of an automated system of traffic control, etc. A significant component in the total cost of ropeways is the cost of manufacturing and installation of intermediate towers along the ropeway line, purchase of traction and carrying steel ropes and of technological equipment.
As shown in [2,23], the task of ropeways intermediate towers construction is a complex task of technical and economic optimization. The purpose of the optimization is to ensure the minimum cost of intermediate towers construction, purchase traction and carrying ropes, as well as a set of technological equipment to be installed on the towers. The setting and solution of this optimization problem allow for a significant reduction of passenger aerial ropeways costs in an urbanized environment [22,23].
One approach to the analysis of the economic component of the problem of urban public transport system modernization based on passenger aerial mechatronic ropeways and to the development of corresponding optimization mathematical models was proposed in [2,25]. Figure 1 shows a design of an aerial ropeways section between two adjacent intermediate towers. Two independent values were proposed to be used as variable optimization parameters of ropeways intermediate towers installation step: towers step t L and carrying ropes tension force k T . Based on them, a vector of controlled parameters [2] was developed: Other technical and economic characteristics of ropeways are proposed to be regarded as fixed: they are either specified as input data or calculated depending on the specified managed parameters. The first group includes [2,25] It was proposed to form a vector of uncontrollable parameters based on the values in the second group. These parameters are not subject to variation in the process of the optimization problem solving: As a result, the task of complex technical and economic optimization of aerial ropeways intermediate towers installation step is reduced to the minimization of the objective function -the total cost of manufacturing and installation of towers, purchase of traction and carrying ropes. According to [2,26], the objective function is written as: And the following should be valid -the restrictions in the form of inequalities [2] which determine the requirements for: -allowable range of change of step between adjacent towers -allowable ranges of traction and carrying ropes diameters -maximum allowable carrying rope sagging between the towers 0 8

PROCEDURE FOR REDUCING OPTIMIZATION CALCULATION LABOR INTENSITY
In [2,26] the solution of the specified optimization problem (3) was performed taking into account the above constraints on variables -distance between intermediate towers t L and tension force of carrying ropes This fact leads to the conclusion that the position of the minimum point of the objective function (3) can be searched not in the entire space, but only along the constraint line (8). Given that the constraint line minimum point location (8) is determined by we can immediately determine the optimal value of one of the variable parameters of the optimization problem: Finding the minimum of the objective function (13) of one variable 1 x can be performed by one of numerical methods of unconditional optimization [27].
There can be a different approach to finding  (13), the following condition is valid opt t opt L x = 1 we get a nonlinear algebraic equation written as follows to determine the desired value: Equation (17) can be solved using one of the numerical methods of solving algebraic equations [28].
If the determined value is there will be two passenger cabins between the adjacent intermediate towers within one span. Therefore, it is necessary to conduct a new iteration of determining the optimal step value using in the specified equations the value cab n = 2.
Similarly, if the determined value is -carrying rope length within one span -carrying rope diameter (23)

ANALYSIS OF CALCULATION RESULTS
To calculate the optimal value of carrying ropes tension force, (12) can be written as 2 2 10 This expression shows that at the specified distance between passenger cabins cab L the value opt k T is determined by the intensity of the distributed load on the carrying rope from the weight of the passenger cabin 2 z . The ratio of the passenger cabin weight and the distance between the cabins at the same value 2 z for the two characteristic values of 20 and 40 N/m is shown in Figure 2. To compare the results of calculations carried out by optimizing the objective function of two variables (3) proposed in [26] and the objective function of one variable (13) proposed in this research, these calculations were performed with the same initial data as in [26]. Two variants of the ropeways were considered based on the use of towers similar in cost to MPG500 and MU330 towers, with two carrying and one traction rope according to Russian standard GOST 3079-80 of marking group k G = 2160 MPa. Calculations following the proposed simplified algorithm, showed that the positions of the minimum points of objective functions (13) and (3) exactly coincided. This testifies to the adequacy of the proposed approach to the accelerated optimization assessment of aerial ropes key technical characteristics. Figure 3 shows the diagrams of change of the value of objective functions (3) along the constraint (  L . The explanation for this is as follows: if such intermediate towers are used, it is economically advantageous to install lower towers, even though their spacing should be smaller and the total number within the ropeway should be bigger. In this case, the trend of increasing costs when using higher towers prevails over the other trend -the increasing cost of installing more towers. On the other hand, (MPG500) towers, characterized by a relatively low rate of growth of their cost with height increase, correspond to a less pronounced extreme form of dependence O(L t ) and large values of intermediate towers optimal step opt t L . In this case, on the contrary, the trend of increasing cost of installing more towers prevails over the other trend -the increasing cost of using higher towers.

CONCLUSION
The method of accelerated optimization estimation of key technical characteristics of aerial ropeways (installation step and intermediate tower height, carrying ropes tension forces) considered in the article allows to obtain exactly the same values as the result of solving the general problem of technical and economic optimization formulated in [2,26]. It has increased computational efficiency, as it is based on the use of simpler mathematical methods, requires less time and computational resources for carrying out optimization calculations.
The proposed method can be recommended for the initial stages of aerial ropeways project development. It allows to quickly determine the optimal values of key technical characteristics of the designed ropeways for a large number of possible combinations of intermediate towers designs, types and dimensions of carrying ropes, cost of technological equipment. Comparative analysis of the results of these calculations provides for the development of an optimally reasoned design solution.