Improving the criteria for choosing the strategies in management by geotechnical systems

The conclusion has been made about the necessity to choose the optimal strategies for management by geotechnical systems, based on the analysis of geological faults, which are the main indicator of the mining and geological conditions that characterize the mineral deposits, as well as on the parameters for the infrastructure development of the underground space. The methodological peculiarity of solving the problems set is the use of game theory with modified criteria of Wald, maximax and Savage, since the manifestation of specific geological faults is probabilistic in nature. When choosing the optimal strategy, the average linear deviations of gains or risks are taken into account.

It should be noted that the most widespread is the morphological classification of dislocations with a break in continuity based on the characteristics: the direction of the relative displacement of the fault wings, the direction and incidence angle of the fault plane. According to these signs, the following groups of discontinuity structures are distinguished: normal faults, expansions, reverse faults, thrusts and lags. This classification divides the faults into those formed under conditions of compression (reverse faults, thrusts, lags) or of stretching (normal faults, expansions) of the earth's crust.
The effectiveness of accepted technological decisions, depending on the probability of a specific geological fault, determines the choice of the optimal strategy for management by geotechnical systems. And this is reflected in the elements of a certain matrix, the interpretation of which is possible based on the methodology of game theory.
The necessity to make decisions based on game theory is determined by a set of criteria and well-known tooling when planning the mining operations and managing the technological processes in mining enterprises [2 -8].

Main part
Let us consider the problem of decision-making under the conditions of indeterminacy [5,[11][12][13]. The indeterminacy means the absence or lack of information about the possible manifestation of geological faults [9]. In this case, it is advisable to use a mathematical model 'Game with Nature'. In the game, a conscious 'player' is a decision-making person (DMP), and 'nature' is a geotechnical system (natural environment). At the same time there is no information about the probabilities of the state of nature.
It is assumed that the player has m of strategies (А 1 ,…, А т ), and the nature has n of possible states (П 1 ,…, П п ). The 'gains' а ij are known of a player at i of a strategy and j of a state of nature. A set of elements а ij form a matrix of game A with size m×n.
In order to choose the optimal strategy, the following classical criteria are used: maximin Wald, maximax and Savage. However, they do not take into account the distribution of all the gains and risks in the different states of nature.
The objective of work is to improve the criteria by taking into account the influence of possible states of nature -geological faults on the choice of management strategy.

Wald criterion
The Wald criterion (criterion of guaranteed result, criterion of pessimism) is one of the most important criteria for reliable decision-making [5,[11][12][13]. It is also used in a number of composite criteria: Hurwitz, Hodges-Lehmann and Germeyer-Hurwitz.
The application area of Wald criterion:  nothing is known about the probabilities of states of nature;  risk is not allowable;  only a small number of solutions is realizable. The Wald criterion is calculated by the formula For each strategy i A , we perform a repositioning (ranking) of gains ij а , arranging them in a non-decreasing order [11,12]. The elements of the obtained matrix are denoted by ij b , and the matrix itself by В . Thus, in each i -th row of the matrix В we have: (1) By virtue of the ratio (1), the first column consists of the minimum gains Hence, the Wald criterion The strategy for which the Wald criterion is valid, will be denoted by W A . In addition to the traditional approach, we will introduce into consideration the allowable interval for the Wald criterion The averaged increment of gain for i -th strategy is characterized by the average linear deviation relative to 1 When comparing the two strategies, the strategy is more favourable that provides the higher averaged increment.

Maximax criterion
The maximax criterion (optimism criterion) is calculated according to the dependence It follows from the ratio (1) that the last column in the matrix B consists of the maximum gains. In this regard, the criterion M can be written in this form The averaged decrease in gain for the i -th strategy will be described by means of the average linear deviation relative to im b Of the two strategies, preference should be given to that one, which has a lower value of i  .
To the right side of (3) we add and subtract 1 i b , and then having performed groupingwe obtain     , we will obtain the dependence between the average linear The formula (5)  .
The first column of the matrix В consists of minimal elements. The maximum of them is (underlined by a single line), therefore, by Wald criterion, the third strategy is optimal. Suppose that for DMP it is permissible to reduce the gain by 0.5. Since 6.8 is in the interval of 6.5-7, then we will compare the strategies 2 A and 3 A according to the average linear increments. The calculations by the formula (2)  .
The strategy should be chosen that provides a greater decrease in risk.
Example. The matrix of gains is given According to the Savage criterion (7), which is equal to 2.8 (it is in bold in the matrix), the first strategy should be chosen. Let the allowable increase in the risk is by 0.5. This condition is satisfied by the second strategy. The calculations by (9)  , then it is rational to choose the second strategy.