Network model optimization of existing mining enterprises to sustain production capacity

The paper is devoted to the sustainability of the production capacities of mining enterprises when extracting minerals. The timely capacity restoration is the main problem for scientists dealing with sustainable development in mining construction. The paper presents the use of network planning and optimization methods of dynamic programming to solve problems of sustaining the existing capacities of mining enterprises. The proposed methods allow for the integrated use of network planning and dynamic programming to optimize the parameters of mining operations. A network model of the technological scheme of mining operations with optimal parameters is proposed. The necessity of timely restoration of capacities of mining enterprises with the use of network planning and mathematical programming methods in solving these problems is highlighted.


Introduction
The mining enterprise's capacity is restored by permanently preparing new working faces to replace the ones that have been exhausted.The lifetime of existing working faces is strictly regulated by the amount of reserves and the daily load on the working face.Its value is calculated by the following formula: n is the number of simultaneously opened floors.
Historically, the problem of timely restoration of capacity of a mining enterprise has been the focus of scientists dealing with the management and organization of mine construction [1,2].The problem has become more acute as mines have transitioned to market conditions.This necessitates the search for new approaches to managing the production capacities of operating mines.
Recently, the theory of project management has been used in investment planning.The term "project" refers to a set of activities limited by a start-to-finish time interval that is designed to create unique products and services.
There are several areas of project management.Among them, the models of calendar and network planning based on the analysis and synthesis of network planning and mathematical modeling methods are of particular interest.
Optimization methods for solving problems aimed at restoring the production capacity of existing mining enterprises are based on the integrated use of network planning methods and dynamic programming.
The cost of restoring the capacity of a mining enterprise accounts for more than 50% of the construction of mine workings: the construction and deepening of shafts, crosscuts, dumping and ventilation drifts, and other mine workings [3,4].The number and length of these workings determine the technological schemes of preparation and exploitation systems in each specific project.
The analysis of the components of the technological scheme allows the organization of mining operations to be represented in a network model.Figure 1 shows a simplified network model of the most commonly used technological scheme for the restoration of working faces, the time parameters of which are calculated according to known methods.The key parameters of the network model are duration and timing of work, critical path, early and late dates of events.
The time parameters of the network model are calculated using well-known formulas [5,6,7].The critical path and its length are determined by events that have an early deadline that corresponds to a late deadline, i.e., events with zero-time reserve.The length of the critical path is determined by the duration of the activities that lie on the critical path.
The network planning methods should be applied to timely solve the problem of building new horizons instead of exhausted ones.For this reason, developing and implementing a methodological approach to the design and optimization of a network model for planning the construction of a new horizon are necessary.
The purpose of the present research is to analyze and prove the use of optimization methods for sustaining the production capacities of existing mining enterprises on the basis of network planning and dynamic programming.

Methods
The network model of mining operations can be viewed not as a program of production processes, but as an innovative project that is optimized according to the criterion of minimum capital expenditures.
The main difference between a project and a program is that a project has a time frame -the beginning and end of the work, a strict time limit and is determined by the length of the critical path.
An innovative mine rehabilitation project is considered completed when the cash flows cease and raw material extraction from the prepared working faces begins [8].
Dynamic programming can be used as a method of optimizing a network model according to the "minimum cost" criterion for a given project duration.
In this case, the criterion equation is described by the equation:

 
).The mathematical model of the problem can be represented as follows: ' '' 1: where i  , i  , i  are coefficients determined using regression analysis.
The resulting problem is a nonlinear programming problem in which the objective function ( 3) is the sum of the capital costs of all operations.The expression in parentheses is the coefficient of increase in estimated costs, which takes into account the transition to higher than standard speeds.Limitation (4) takes into account the need to complete the critical path by the directive deadline 0 T .The technological limitations on the sinking rate of the i -working face are taken into account by inequalities (5) [9].
Problem ( 3) -( 5) can be solved by dynamic programming.It is more convenient to move to new variables, so that the unknowns are the durations of the relevant operations.
( ) Denote by ( ) ii kt the costs for mining the -working face, which depend on the duration of mining operations.
From expressions (3) and ( 6) it follows that ( ) ( ) Taking into account the new designations, the task can be rewritten as follows: ( ) where ( ) Consider m -step management process in which each subsequent step involves the selection of the next working face on the critical path.The process at each step can be described by the time T , months, remaining until the completion of the entire complex of works before the directive deadline 0 T , months, therefore, T is the phase coordinate.The management at each step is the found value of t the time of production, months.The efficiency criterion is the total capital expenditure K , cur.
Obviously, the efficiency criterion is adaptive, and the new process is determined by its previous state and the adopted management Denote by n the ordinal number of the steps in the reverse calculation, and write the recurrence relation Here ( ) n fT is the minimum cost of final mining the working faces, provided that before the directive term 0 T there are T months left, monetary units.The phase coordinate T must satisfy the following conditions ( ) This means that for all steps except the first one, the time remaining before the final event (the startup of the first stage) cannot be less than the minimum total duration of subsequent work and greater than the remainder of the directive period 0 T , which is formed when performing previous work with a minimum duration [10].At the first step, the phase coordinate is set to 0 TT − .Possible managements t at each step are determined by the conditions The left inequality shows that the duration of the respective operation cannot be less than the lower technologically permissible limit.The right inequality takes into account that the management cannot exceed the upper technologically permissible limit and the time remaining before the directive deadline when carrying out the following operations in the minimum time.
Using the recurrent relation (13), the conditionally optimal managements ( ) ( ) Then, based on the initial state of the process (the directive term 0 T ), the optimal managements * i t are determined at each step.
After optimizing the critical path operations, the network graph should be analyzed again.If the calculations show that a new critical path has appeared, the work of which cannot be completed by the directive deadline 0 T , then the duration of the new critical path is similarly optimized [11].

Results and discussion
During the construction of a mine, the directive time for starting the first stage is 30 months.Network graph calculations have shown that there are four mine workings on the critical path (cage shaft, haulage crosscut, main face, and brake incline).The relevant characteristics of these workings are given in table 1, which shows that at the normative sinking rates, the duration of mining operations was 40 months, which exceeds the directive period by 10 months.The requirement for solving the problem is to increase the sinking rate within technologically acceptable limits so that the work can be completed by the directive deadline at the lowest possible cost.
The coefficients required to determine the technologically permissible limits of sinking rates, as well as the regression coefficients, are shown in table 2, which shows that the normative sinking rates are taken as the lower limit.The values we are looking for are the duration of the mining workings.Since knowing them, the rates can be determined by the formulas (6).
The limits of change in the duration of mining workings within technologically permissible limits, determined by formulas (11), are given in table 3. When optimizing by dynamic programming, we take the step of varying the desired values equal to 1 month [12].
Consider the optimization at the last step ( ) We apply the recurrence relation (13) when 1 n = .
( ) ( ) ( ) The phase coordinate T must satisfy the inequality 5 T  , otherwise it is impossible to complete the first stage of construction by the directive deadline with the technologically permissible duration of the fourth working.
On the other hand, 17 T  , since 17 is the maximum amount of time left before the directive deadline 0 .The right-hand side of the inequality means that the duration of the last mining working cannot exceed the upper technological limit of 10, as well as the time T , remaining before the directive deadline 0 T .This corresponds to inequalities (15).
The calculation of ( ) kt are carried out according to formula (7), the numerical values of the coefficients are taken from tables 1, 2. The function ( ) The calculation of ( ) fT and the conditionally optimal management ( ) The choice of values T and t is determined by the above inequalities.At the intersection of the corresponding rows and columns, the values ( ) ( ) ( ) ( ) ( ) The phase coordinate must satisfy the inequality 11 T  , which means that at the beginning of the penultimate step until the directive deadline 0 T there must be a time not less than the total time of the last two workings at maximum speeds.On the other hand, 23 T  is the maximum time remaining before the directive deadline 0 T , if the first two workings are completed in the minimum time ( ) Therefore, 11 23 T  , which corresponds to inequalities (14).The range of admissible managements is determined by the inequalities ( ) The right-hand side inequality means that the duration of the third mining working (main drift) cannot exceed the upper technological limit of 12, as well as the time 5 T = , which remains if the fourth working (brake incline) is carried out in the shortest time (5 months).
The calculation of ( ) fT and the conditionally optimal management ( ) uT is similar to that in table 4. The choice of values of T and t is determined by the above inequalities.At the intersection of The minimum value of this sum, i.e., ( ) fT, for each row is written in the penultimate column, and the value of t , at which this minimum is reached is placed in the column ( ) Cells with invalid combinations of T and t are not filled.The function ( ) kt is calculated using formula (7), with the coefficients taken from tables 1 and 2. The values of: ( ) Consider optimizing the time of the second working face ( ) Similarly to the previous calculation, we find that the phase coordinate T must satisfy the inequality The calculations are presented in table 5. Now consider the process from the first step to the last.The optimal duration of the first mining working is shown in table 5. Find the process at the end of the first step.According to the table of the third step of the calculations, we determine the optimal duration of the second mining working.
Next, we calculate similarly using

T T t t u T u
Thus, the optimal duration of mining operations is 12, 4, 8, 6 months, respectively.The total capital costs are equal to the minimum possible value of ( ) The optimal management of the complex of mining operations on the critical path is presented in table 6.Comparison of the obtained optimal rates in the table 1 shows that only the first working face should be mined at the normative rate, while the other three rates should be higher than the normative ones.
Note that the capital costs at the normative sinking rates are as follows ( ) ( ) ( ) ( ) This means that in order to complete the mining operations on the critical path by the directive deadline T0, it is necessary to increase the sinking rates of the second, third and fourth workings compared to the normative rates.This increases the cost of the mining workings by UAH 522.06 thousand, i.e. 2.8% of the estimated cost.
The accuracy of the outcomes varies with the selected value step.It increases with a smaller value step.At the same time, the amount of calculation increases.

Conclusions
The use of optimization methods for sustaining the production capacities of existing mining enterprises based on network planning and dynamic programming improves the management of operations in accordance with the deadlines.The optimization in the example above aims to reduce the length of the critical path required for the construction of a new horizon.
The optimized construction of the new horizon resulted in a 25 % reduction in construction time compared to the initial data.
Thus, this paper focuses on the relevant aspects of developing and implementing a methodologicalbased approach to creating and optimizing a network model aimed at sustaining the production capacities of existing mining enterprises.
In summary, this paper shows the main advantage of mathematical modelling of the network schedule aimed at implementing a new construction process over traditional practices by providing a methodology for achieving the goals set in a particular construction project according to the selected quality criteria.

h
is the height of the working floor, m; 0 v is the annual reduction of second workings, m/year;
written.Each row contains the minimum value of this quantity ( )1fT, and the value at which this minimum is reached is placed in the column ( )1uT.Cells corresponding to invalid combinations of T and t , are not filled.Note that when 10 17 T  the minimum is achieved with the same management ( ) optimize the penultimate step 4 i = .We apply the recurrence relation (13) at 2 n = : similarly to the previous steps.Time of the first mining working ( )1 i = isoptimized in the same way.The only simplification is that the phase coordinate can acquire only one value 30 T = , and the management t must satisfy the conditions

Table 1 .
Comparison of optimal rates of mine workings construction.

Table 2 .
Technological limits of sinking rates and corresponding regression coefficients.

Table 3 .
Determined permissible technological limits for the mining workings' duration.

Table 4 .
The results of the first pass in determining the optimal duration management, months.

Table 5 .
Optimal duration of the first mining working, months.
table 4 and the second step of the calculation

Table 6 .
The optimal management of the complex of mining operations on the critical path.