MODELING OF OPERATING PROCEDURES IN A FOREST COMPARTMENT UNDER FUZZY DYNAMIC NATURE WORKING CONDITIONS

In this paper the principles and calculations of stretched in time graphic-analytical models of the operating network are proposed. They allow you to determine the sequence of logging opera-tions at logging sites within the forest compartment, to reduce nonmechanical delay and increase technological effi ciency of harvesting operations with fuzzy dynamic natural-production conditions. Search algorithm minimal cost fl ow developed to solve the proposed graphic-analytical model. It allows to justify the amount of expenses and calculate a maximum bandwidth of arcs between the fi ctitious source and drain of graphical model. The model differs complex use of two main charac-teristics of the process of logging operations: productivity and labor costs are presented as individu-al capacities of graph arcs. In order to solve the suggested graphic model, we developed the search algorithm of mini-mum cost fl ux, providing grounds for the typical expenditures and calculating the carrying capacity between the false source and runoff within the structure of suggested graphic models, different in two core characteristics of the logging process: productivity and labour inputs, represented as the carrying capacity of the graph arcs. The suggested graphical-analytic approach represents a certain task type of graph theory, according to which the transportation of fl ux along the dynamic opera-tional network arcs reduces their carrying capacity. Every arc denotes analogous technological op-erations within the analysed time period. This is the distinctive feature of the suggested approach. The suggested design and calculation principles for time expanded graphical-analytical models of the operational network enable to provide rationale for the logging consequence and technological operations in fuzzy dynamic nature environment of a forest compartment, reduce operational downtime and enhance the effi ciency of forest harvesting operations.


INTRODUCTION
It is well-known that selecting a set of machines when forming a park of timber harvesting equipment at a forest engineering enterprise should be based on the approach when the effective average of all the machines included into a system equals or divisible by the productivity of the leading machine. However, the nature working conditions in every harvesting area are signifi cantly different from the analogous mean values within the enterprise. Any deviation of the nature working conditions in every harvesting area from the main values, characterizing the total felling-area resources of the enterprise result in inconsistent operation of the whole harvesting system. This problem may be partially mitigated, when a few areas are harvested in one forest compartment [05,06,07,08,09,10] based on sequencing feasibility and harvesting technique applied to each of them. It should be taken into consideration, that in real working environment of harvesting areas such parameters as production of harvesting machines and prime costs of transportation and timber processing operations in ever-changing conditions of the environment can't be certain in advance. Presentation of the technological processes sequence in the forest compartment in the form of a graphic-analytical model brings a researcher to a time independent and dynamic fl ow problem in fuzzy dynamic nature working conditions of the environment, which doesn't presume instant pass of the whole fl ow on the arcs of a graph. The research aims to provide rationale to the operating procedures carried out in a forest compartment with integrated account taken of special factors relating to technological characteristics of the all the harvesting areas within the compartment.

LITERATURE REVIEW
Flow optimization in transportation is revealed in [01, 02, 03]. In [04] the author suggests a solution to the problem of the maximum fl ow, in which the transportation parameters are regarded as fuzzy triangular numbers, the addition, subtraction and comparison tasks for fuzzy triangular numbers are solved. The research described above gives an idea on the possibilities of making calculations in fuzzy conditions of a transportation network. However, this research cannot be applied to managing the operating procedures of wood cutting operations without due consideration of the sector-specifi c issues.

MATERIALS AND METHODS
In order to achieve the set goal, we developed the following graph options used to carry out all kinds of technological processes of harvesting operations. For example, in Fig.1 there is a time expanded dynamic graph, characterizing a technological process of forest compartment felling including all the processing operations of a technological process on a forest swath. The graph is drawn by the formation of a separate copy of every vertex in every period under review is represented by arcs, formed by every couple "vertex-time" (x i ,θ) X p into every couple "vertex-time" (x j ,θ) and . Thus, x j Г(x i ), and . The transportation capacity of , connecting the couple "vertex-time" (x i ,θ) and equal ∞, while the transportation capacity of , connecting the couples "vertex-time" (x i ,θ) and (x j ,θ) equal and can be calculated by means of the analysis of man-hour valued defi ned in the graph used to carry out certain operations. Vertex S represents a false source, while vertex T represents a false runoff. Every process procedure is represented in the graph as intermediate vertex , located between a source and a runoff. Where N-is the name of the process operation, h -is the sequence number of an operation in the overall sequence of the process fl ow in a forest compartment; l -is a number of the analyzed harvesting area, θ -is the number of the analyzed time period. Thus, for example, vertices refer to the processing operations (felling, limbing, bucking), carried out on the forest swaths of every of g analysed harvesting areas for the fi rst time period, vertices refer to a skid logging transportation to the loading point carried out within a second time period, vertices refer to an operation of timber hauling from the loading point in the fi rst time period, etc. The transportation capacity of every false arch originating from a source is limited by the deposit of harvested timber on every of g wood-cutting area located within a forest compartment. This indicator on a graph may be represented taking into consideration fuzzy transportation capacity of arcs, and be denoted by Since this source is a false vertex, then the cost parameters of timber transportation along the arc going out equal 0. The arcs of this graph drawn between the vertices, indicating the operations of one time period, characterize the movement of raw timber on the stages of technological process. The fl ow of the process along these arcs is characterized by fuzzy fl ow capacity of arcs, denoting the productivity of harvesting machines and equipment, which are in direct relations to labour inputs necessary to complete every operation, as well as fuzzy variable costs . The latter include fuel cost, piecework wages, etc. Fuzzy variable costs are fi xed when fl ows move between the operations of a technological process. The movement cost of fl ow unit along the graph arcs connecting the time-vertex couple (x i ,θ) and (x j ,θ) equal . Thus, it is possible to take into consideration both fi xed and variable costs necessary to carry out operations analyzed in a graph. The arcs drawn between the vertices referring to the same operation but carried out in different time periods, refer to the raw timber somehow excluded from the subsequent state of the technological process in the time periods considered earlier. Thus, for example, fi xed costs (Z), which are fi xed when the fl ow moves from one time period into the next, include depreciation costs, planned maintenance and machine repair costs, wages costs provided that employees are paid based on hourly basis, etc. The movement cost of fl ow unit along the graph arcs connecting the time-vertex couple (x i ,θ) and equal Z ij (θ). The problem of a maximum fl ow at a minimum cost in the suggested fuzzy dynamic network of process operations of timber harvesting in a forest compartment may be formulated using the following mathematical relations: 1. It is necessary to defi ne the minimum route of timber movement along the arcs of a dynamic operational network in the suggested number of periods. This equation should be true for all the time periods and all graph vertices, besides the source and the arcs, connecting different time periods.
where b-is the ordinal number of the analysed arc (1≤b≤g); m-is the maximum working time of the analysed period measured in man-hours; -is labour inputs used to process or transport a timber unit when carrying out N -operation of the technological process in θ -time period, h/m 3 ; -is the fl ow value, transported from vertex i into θ -is the time period for carrying out N -operations of the technological process, m 3 . One of the parameters, restricting the capacity of arcs is the productivity П(θ). In our case by productivity we mean the scope of work, which can be completed before the end of the analyzed time period. In the course of calculation, as a result of cargo traffi c movement along the graph arcs, there is a reduction of time m*(θ) left before the completion of the analysed stage, which equals: Thus, the labour inputs used to carry out this or that operation of a technological process and machine and mechanisms performance are interconnected: Under fuzzy natural environment the productivity and labour inputs may be represented as fuzzy triangular numbers: The initial values of the left and right borders of fuzzy triangular numbers are set by the experts. These indicators are interrelated and can be calculated: The formulas presented make is possible to get the inverse relation: The labour inputs, characterising the inverse arcs equal as follows: Where is the time of technological operation, which forms the inverse arc on the graph. The limiting values of the triangular numbers, denoting labour inputs may be calculated based on the relations.
A detailed numerical example of solving the problem is considered in [10].

RESULTS AND DISCUSSIONS
In order to solve the suggested graphic model we developed the search algorithm of minimum cost fl ux, providing grounds for the typical expenditures and calculating the carrying capacity between the false source and runoff within the structure of suggested graphic models, different in two core characteristics of the logging process: productivity and labour inputs, represented as the carrying capacity of the graph arcs. The suggested graphical-analytic approach represents a certain task type of graph theory, according to which the transportation of fl ux along the dynamic operational network arcs reduces their carrying capacity. Every arc denotes analogous technological operations within the analysed time period.