Determination of optimal values for the technical characteristics of construction machinery unit with application of fractional linear programming

. Paper covers issues connected to the determination of the technical characteristics of construction machinery unit performing a certain category of work. On the basis of the results of the review and analysis of scientific works concerning the problem under consideration the key disadvantages have been identified for the existing techniques. Optimization model for determination of the technical characteristics of construction machinery unit has been created. Initial form of the model is not linear toward unknown variables but it can be transformed to the linear form that allows application of simplex method for efficient solution of the problem. The created model has been implemented on a practical case – the obtained results indicated the adequacy of the developed tool.


Introduction
Modern conditions of the construction industry's development are characterized by the large variety of corresponding machinery models with different technical characteristics and, as sequence, different rental costs and productivity indicators. That circumstance determines the high relevance of the problems connected to formation of optimal structure for construction machinery fleet on the basis of time and cost criteria in conditions of complex structure of the construction projects and large amount of variants for machines' design and productivity. The relatively sophisticated approach is based on the use of the modern simulation and optimization techniques that are based on mixed-integer linear programming [1][2][3], particle swarm optimization [4], ant colony optimization [5], genetic algorithms [6], etc. The corresponding scientific developments assume creation and implementation of complex mathematical models and algorithms to take into account the large number of important factors that have influence on the machinery fleet's efficiency indicators and obtain adequate results. Still, the corresponding calculations' conduction process require high qualification in the field of mathematics and programming for the person solving the task. The more simple (in terms of practical application) approach is based on the ranking of the available machinery variants (models) connected to the certain category of work with further selection of the appropriate number of the most preferred machinery models (in accordance with the rank and available number) to provide the required productivity of construction machinery complex for each category of work. To calculate the rank for machinery variants under consideration, it's necessary to justify the composition and values of the reference characteristics for the machinery unit -it will allow to calculate the deviations between actual and reference values of characteristics for each machinery model and determine the rank in accordance with the following principle: the less the value of generalized deviation for machinery model, the bigger the value of corresponding rank. However, the preliminary analysis of the scientific works connected to the justification of the machinery unit's characteristics showed the lack of effective tools providing calculation of reference values for machinery characteristics with full and objective taking into account of cost and productivity indicators. On the basis of the features indicated above the conclusion has been made about the necessity for conduction of the research aimed to the creation of the tools for the determination of the reference technical characteristics of construction machinery unit. The main tasks of the research have been formulated as follows: 1. Carrying out of the review and analysis of the modern scientific developments in the research field. 2. Creation of the optimization model for determination of optimal values for the technical characteristics of construction machinery unit with application of fractional linear programming. 3. Implementation of the created model on a practical example.
The object of the research is construction technological process that assumes performing of the certain category of work by cyclic-operating machinery unit within limited time interval. The subject of the research is the characteristics of the mentioned above process in terms of technical and cost characteristics of the machinery unit.
At the initial stages of the research the review and analysis of the scientific developments [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] in the field of determination of characteristics for the machinery unit within the technological (particularly, construction) process. Most scientific developments assume determination of the machinery's characteristics by selection of the most appropriate variant through the application of hierarchy analysis methods [7][8][9][10][11][12][13][14][15][16], fuzzy logic techniques [17,18] and their combination [19]. The common disadvantage of the corresponding procedures is connected to the direct dependence of the solution search's laboriousness and the number of alternative variants of machinery. Additional disadvantages of the mentioned above categories of scientific developments are the following: dependence of the results on the expert opinion with a certain degree of subjectivity; the difficulty of providing an objective assessment of the objects of analysis [7][8][9][10][11][12][13][14][15][16]19]; the difficulty of objective description of the interconnection between different categories of parameters (including qualitative and quantitative) [17,18]. Mentioned above conclusions determined the necessity for creation of the optimization model which would provide determination of the technical characteristics of cyclicoperating construction machinery unit with adequate and objective account of the criteria of its productivity and economic efficiency. More detailed information is presented in the next sections of the work.

Methods
At the intermediate stage of the optimization model has been created on the basis of preliminarily formulated main provisions and supposed composition of initial data and

Main provisions
Main provisions of the elaborated model are the following: the object under consideration is the technological process connected to the handling of cargo units (CU) by a unit of construction handling machinery (CHM) of cyclic action with battery power supply within the corresponding technological cycles, implemented within a limited time interval. the technological cycles implemented by the CHM unit correspond to a certain category of work and therefore have similar configuration parameters of the corresponding technological routes. the similarity of the configuration of technological routes implemented by the CHM unit within the corresponding cycles of CU processing makes it possible to form the most probable (median) technological route, the implementation of which within the considered time interval will ensure the productivity of the CHM unit, equal to the actual productivity of the CHM during the implementation of technological routes with similar, but not identical configuration. the general composition of the CHM technical characteristics includes the main technical characteristics that directly determine the rental cost for a CHM unit, and additional characteristics that do not significantly affect the rental cost. The list of the main characteristics for the considered CHM is presented in Table 1.  MSU denotes a variable measure unit, the description of which is presented in Table 1); the configuration of the technological route describes the trajectory of the spatial movement of the CHM unit during the implementation of technological cycle and includes straight and curved (turns) sections; the passage of straight sections is carried out by a CHM unit at a stationary travel speed (for example, in the process of CU's transportation from the place of pickup to the place of release) or at a shunting travel speed (during operations of CU pickup or release). the passage of curved sections is carried out by a CHM unit only at shunting speed. there is a certain relationship between the travel speed of a CHM unit without CU and with it, the same for both stationary and shunting speeds. there is a certain ratio between the stationary and shunting travel speeds of a CHM unit, which is the same for the movement both with and without CU. there is a certain ratio between the speeds of the vertical movement of a CHM unit's pickup device with CU and without it, the same for the pickup device's lifting and lowering. there is a certain ratio between the speeds of lowering and lifting the load gripper, which is the same for the pickup device's movement both with and without CU. there is a certain ratio between the total mass of a CHM unit and its load carrying capacity. there is a certain ratio between the mass of a CHM unit's pickup device and its load carrying capacity. it is necessary to determine the values of the CHM technical characteristics, at which the maximum value of money saving for the rental of the CHM unit, referred to the average duration of the implemented technological cycle, is ensured.

Model description
Source data and unknown variables connected to the created optimization model are presented in Table 2 and Table 3 respectively. Calculated characteristics used within the model and independent of its unknown variables, are presented in Table 4.   (1) the measure unit indicated for certain elements of source data and specified as MSU has a different meaning depending on the index of CHM technical characteristic (for more information, see Table 1); (2) Value of source data element must correspond to the condition also must take into account features of the technological process's implementation -for example, CHM load carrying capacity c G must be equal or higher the CU mass m , turning radius trn R must be close to the corresponding calculated value of technological route R , etc. Table 3. Unknown variables for the created optimization model.

No.
Name of the unknown variables' category Meas. unit (1) Designation / expression 1 Unknown variables of the base model (original unknown variables) 1.1 Value of CHM technical characteristic (2)  (1) the measure unit indicated for certain categories of unknown variables and specified as MSU has a different meaning depending on the index of CHM technical characteristic (for more information, see Table 1); (2) alternative designations of unknown variables ^ì x in expressions in items 1.1 and 2.2 are presented in Table 1.

Results and discussion
At the final stage of the research created model has been implemented on the practical case -determination of the optimal values for the technical characteristics of the reach-trucks operating within the storage zone for construction materials. Preparation of the model was carried out in "Microsoft Excel" software, implementation of the model was conducted with the use of "Solver" add-in. The general view of the "Microsoft Excel" worksheet is shown in Fig. 1. The settings of "Solver" add-in are presented in Table 5.  As received results correspond to the assigned ranges of permissible values, the created model is adequate and can be used as the effective tool for solving of corresponding applied tasks. However, there are several important features that have significant influence on the practical significance of the optimization results. The mentioned-above features are the following: adequacy of the optimization results is directly connected to the adequacy of the analytical model describing the dependence of the rental cost for a CHM unit on its technical characteristics (see expression (1)). So, before the preparation of the optimization model, the adequacy of analytical model must be verified (for example, with the use of determination ratio 2 R ); configuration of the feasible region (and optimal solution as well) depends on the value of parameter ASB G (item 3.3 in Table 2) within the element min cond ASB A (item 2.5 in Table   4) of the expressions (12) and (13) connected to the indirect constraints. Nevertheless, as the parameter ASB G varies from 0 to 1, it is possible to find the single value of parameter ASB G that corresponds to the most preferable variant of optimal solution of the model by sequential conduction of optimization experiments with different values of parameter ASB G .

Conclusion
On the basis of the conducted research, the following main conclusions have been made: 1. Review and analysis of the modern scientific developments in the research filed has been carried out. Corresponding results have shown the lack of the tools for determination of the construction machinery unit's characteristics that provide adequate taking into account of productivity and economic efficiency indicators. 2. Optimization model for determination of the construction machinery unit's technical characteristics with application of fractional linear programming has been created. On the basis of the results of the model's implementation on a practical case the conclusion has been made about the adequacy of the proposed tool.