$title Opencast Mining (MINE,SEQ=39)

$onText
This model finds an optimal extraction schedule for an opencast mine
with a side angle of 45 degrees and square plots. The extraction blocks
are identified by level, row and column number, with the surface blocks
having level number one.

Williams, H P, Model Building in Mathematical Programming. John Wiley
and Sons, 1978.

Keywords: linear programming, mining, scheduling
$offText

Set l 'identifiers for level row and column labels' / 1*4 /;

Alias (l,i,j);

Table conc(l,i,j) 'estimated ore concentration (percent metal)'
           1    2    3    4
   1.1   1.5  1.5  1.5  .75
   1.2   1.5  2.0  1.5  .75
   1.3   1.0  1.0  .75  .50
   1.4   .75  .75  .50  .25
   2.1     4    4    2
   2.2     3    3    1
   2.3     2    2   .5
   3.1    12    6
   3.2     5    4
   4.1     6               ;

Set
   k        'location of four neighboring blocks' / nw, "ne", se, sw /
   c(l,i,j) 'neighboring blocks related to extraction feasibility'
   d(l,i,j) 'complete set of block identifiers';

Parameter
   li(k)    'lead for i'  / (se,sw)   1 /
   lj(k)    'lead for j'  / ("ne",se) 1 /
   cost(l)  'block extraction cost' / 1 3000, 2 6000, 3 8000, 4 10000 /;

Scalar value 'extracted block value if 100 percent metal' / 200000 / ;

c(l,i,j) = yes$((ord(l) + ord(i)) <= card(l)     and (ord(l) + ord(j)) <= card(l));
d(l,i,j) = yes$( ord(l) + ord(i)  <= card(l) + 1 and  ord(l) + ord(j)  <= card(l) + 1);

display c, d;

Variable
   x(l,i,j) 'extraction of blocks'
   profit;

Positive Variable x;

Equation
   pr(k,l,i,j) 'precedence relationships'
   def         'profit definition';

def.. profit =e= sum((l,i,j)$d(l,i,j), (conc(l,i,j)*value/100 - cost(l))*x(l,i,j));

pr(k,l+1,i,j)$c(l,i,j).. x(l,i+li(k),j+lj(k)) =g= x(l+1,i,j);

x.up(l,i,j) = 1;

Model mine / all /;

solve mine maximizing profit using lp;

Parameter rep(i,j,l) 'extraction decision table';
rep(i,j,l) = x.l(l,i,j);

display rep;