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76 Advances in Productive, Safe, and Responsible Coal Mining

width of the panel from barrier pillar center to center, ρ i is the width of the barrier pillar
designed for panel width i, and π i is a production rate index for panel width w i , which
is proportional to the estimated production rate for the particular panel width. It is also
useful to ensure a production rate index of similar magnitude to panel width. The deci-
k
sion variable x i is the number of panels of width w i used in strip k. The objective is to
maximize recovery by maximizing the overall width of coal that is mined
π
P K P m k P K P m k
( ð w i ρ Þx ) and overall production rate ( i¼1 i x ):
k¼1 i¼1 i i k¼1 i
K m
XX k
max f η w i ρ Þ + π i gx i
ð
i
k¼1 i¼1
m
X k (5.2)
w i x W k 8k
i
i¼1
k
x is a nonnegative integer
i
For this formulation to work, the user must carefully select values of η and π i . The user
can specify π i as the simulated production rate from the DES model or an index that is
proportional to the production rate. It is important that values of π i are distinct enough
that the solution discriminates between various panel widths under consideration. For
example, a linear scale from 0 to 100 can be used to define values based on production
rates (Eq. 5.3). Variables r i , r min , and r max are the production rate for panel width i, the
minimum production rate, and the maximum production rate in the given set of panel
widths, respectively:
r i r min
π i ¼ 100 (5.3)
r max r min
It is also important to ensure that η is chosen to reflect the relative importance of recov-
ery and production rate to the decision. If management desires for production rate and
recovery to be equally important in the decision, then η should be chosen to ensure that
ηw i is of the same order of magnitude as π i . Otherwise, the user should specify η such
that ηw i ≫π i or vice versa.

5.3.2 Solution formulation

Eq. (5.2) is an integer optimization problem that can be solved with a variety of solu-
tion algorithms including branch and cut and branch and bound [12]. In the case study
described in the next section, an illustrative example of the problem is solved using
CPLEX together with its MATLAB application program interface (API). This allows
preparation of the problem in MATLAB, which is then passed to CPLEX. CPLEX
solves the problem and returns the solution to the MATLAB environment where
results can be postprocessed into useful input for mine design. The CPLEX integer
optimization solver is used, which is based on a branch-and-cut algorithm.
The CPLEX integer optimization solver solves problems like Eq. (5.4). Thus, to
solve a problem using the solver, one needs to convert the problem input into vectors