Page 90 - Advances In Productive, Safe, and Responsible Coal Mining

Optimization of coal recovery and production rate as a function of panel dimensions 75

exist depending on the nature of the problem. There are many variations of the cutting
stock problem with different objectives.
The one-dimensional problem is the most common and exists in multiple industries
including fiber, paper, steel, timber, and aluminum industries. In this problem, a roll of
material needs to be cut into pieces of different lengths (but the same width, which is
the same as the width of the roll) that minimize the trim loss (what is left over after
cutting).
Assuming that the width of paper rolls is W>0 and the customer i (i¼1, 2, …, m)
wants b i cuts of width w i W (orders for cuts of width greater than W cannot be ful-
filled), the maximum number of rolls needed, K, can be estimated by considering that
P m
b
in the worst case, each cut is from one roll (i.e., K ¼ i¼1 i ). The one-dimensional
cutting stock problem can then be formulated as Eq. (5.1):
K
X k
min x
0
k¼1
K
X k
x b i 8i
i
k¼1
(5.1)
m
X k k
w i x Wx 8k
i 0
i¼1
k
x 2 0, 1g 8k
f
0
k
x is a positive integer
i

0, if rollk is not used
k k
where x ¼ and x i is the number of cuts of width w i from
0 1, if roll k is used
roll k.
For large numbers of cuts and stock rolls, the size of the linear programming prob-
lem becomes prohibitively large. Consequently, solutions to the cutting stock problem
are challenging if all basic solutions are examined. Hence, most research on cutting
stock problems has focused on how to solve problems without examining all possible
solutions.
The authors posit that the optimization of coal recovery based on panel widths is a
one-dimensional cutting stock problem. Each strip of the underground coal mine,
which is made up of a series of panels separated by barrier pillars, can be considered
a stock roll. Hence, the whole mine is made up of K stock rolls of differing widths, W k
(k¼1, 2, …, K). Each panel can be w i wide, where i¼1, 2, …, m for the m panel widths
under consideration. Unlike the conventional cutting stock problem, the coal recovery
problem does not have specific demands for how many times a particular panel width
is used in the design. Also, the coal recovery problem requires that all strips be mined.
This is analogous to having all available rolls used in the cutting stock problem.
k
Hence, there is no need for the variable x 0 , which defines whether a roll is used or not.
Therefore, the coal recovery problem can be modeled with Eq. (5.2) where η is a
ratio used to specify the relative importance of recovery and production rate, w i is the