Roof constraints

The roof rotation zone. This zone occurs approximately from 0 m to 20 m in front of the face. In this zone the magnitude and rate of lateral movement of the top coal toward the gob increase rapidly under the pressure of the broken overhung immediate roof Constraints from the gob side reduce greatly, allowing the fractures in the top coal to expand freely. In this zone, the fractures expand further, and top coal loses stability and begins to fall down. 

As previously mentioned, in the Attic method, it is important to discriminate between attic vertices and interior points, between different types of constraints, and between variables with floor or roof constraints. 

In the starting point, the Attic method verifies whether any variable has only floor constraints as satisfied constraints. In this problem, both the variables have the same sign in both the objective function and in the satisfied constraints hence there are no roof constraints for them and it is possible to perform a search along X by obtaining xi = l x2 =0 F = —5, and along x 2 by obtaining xi = 0 x 2 = 1 F = —6. 

There could be 1 to uy roof constraints and their complementary constraints are floor constraints. 

Roof constraints have the peculiarity that each vertex on them is better than the corresponding (complementary) vertex lying on the floor constraint. This is no longer true for wall constraints. 

Each roof constraint is present in 2" vertices and each of these vertices inserted into the matrix J has A > 0 for the roof constraint. 

Figure 10.3 shows two three-dimensional cases with 2 and 3 floor constraints and an even number of roof constraints. 

Figure 10.3 Graphical representation of linear programming with two and three roof constraints.

If one or more constraints progressively inserted are roof constraints, they are not removed anymore from the matrix J as they have k > 0 for all new vertices. 

The number of possible constraints to insert decreases by one for each roof constraint already inserted, whereas the number of vertices that could potentially be the problem solution is halved. 

The worst scenarios for the initial point are when both no variables with floor constraints can be modified to improve the objective function by making some constraints passive and there is a single roof constraint in the problem dimensioned ny. 

After a maximum number of iterations equal to tty, a vertex with a roof constraint is achieved. From this point, the roof constraint is active and inside the matrix J until the solution is reached. The novel problem obtained is of the same class, but reduced by one in the problem dimensions. If the worst-case scenario also occurs in this new problem (i.e., a single roof constraint), a second roof constraint, this time for the subspace Wy — 1, is found after a maximum of wy — 1 iterations. The procedure is iterated until the solution is achieved. At each step, a problem from the same class, but with dimensions reduced by one, is solved in fact, one more roof constraint is inserted. 

The number of iterations can be smaller than the iteration amount gjven in Table 10.1 when there are variables with floor constraints in the initial point as well as when the number of roof constraints is larger than one in some subspaces in which the iterations are performed. For example, if ny roof constraints are present, the number of iterations becomes ny see in Figure 10.3 the sequence vertex 1(------), nonvertex (x + x), nonvertex (x + +), vertex 8(+ + +). 

Note that starting from the vertex with all the constraints — the new vertex with all the constraints + is achieved after wy iterations. The constraint Wy" + is the roof constraint for the problem dimensioned Wy. This constraint is now preserved as an active constraint in the matrix J and the resulting optimization problem decreases by 1 dimension and the new problem is a subspace of wy — 1 dimensions. This new subspace has wy — 1 roof constraints in correspondence with the remaining constraints — wy — 1 iterations are enough to achieve the... 

The reason behind the first point should be clear if a constraint with Aj < 0 is removed, all the constraints subsequently inserted into the matrix are not necessarily roof constraints even though they have A > 0 at the working vertex, since their insertions were conditioned by the existing constraints with Aj < 0 they are, however, replaced by artificial constraints and therefore the number of iterations to achieve a new vertex does not increase. Conversely, the constraints with a matrix row index smaller than the first constraint removed are reasonably... 

When variables with no roof constraints exist in the initial point, the best search performed along these axes is selected and compared to the search in the direction providing the maximum improvement of the objective function. 

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