Opencast Mining: How Much to Excavate


PROC OPTMODEL Statements and Output

The following PROC OPTMODEL statements declare index sets and parameters and then read data sets to populate them:

proc optmodel;
   set BLOCKS;
   num level {BLOCKS};
   num row {BLOCKS};
   num column {BLOCKS};
   num revenue {BLOCKS};
   read data block_data into BLOCKS=[_N_] level row column revenue=percent;
   for {block in BLOCKS} revenue[block] = &full_value * revenue[block] / 100;

   set LEVELS;
   num cost {LEVELS};
   read data level_data into LEVELS=[_N_] cost;

The following statements declare binary decision variables and the objective:

   var Extract {BLOCKS} binary;
   num profit {block in BLOCKS} = revenue[block] - cost[level[block]];
   max TotalProfit = sum {block in BLOCKS} profit[block] * Extract[block];

The following CON statement uses the colon operator (:) to enforce the rule that if you extract block i, then you must also extract each block j that lies above it:

   con Precedence_con {i in BLOCKS, j in BLOCKS:
      level[j] = level[i] - 1
      and row[j] in {row[i],row[i]+1}
      and column[j] in {column[i],column[i]+1}
      }:
      Extract[i] <= Extract[j];

The following SOLVE statement calls the MILP solver. However, since the constraint matrix is totally unimodular, the optimal solution of the LP relaxation is automatically integral, and hence no branching is required.

   solve;
   print Extract profit;

The following CREATE DATA statement uses the colon operator (:) to output only the blocks that are used in the optimal solution:

   create data sol_data from
      [block]={block in BLOCKS: Extract[block].sol > 0.5}
      level row column revenue cost[level[block]] profit;
quit;

Figure 14.2 shows the output from the mixed integer linear programming solver.

Figure 14.2: Output from Mixed Integer Linear Programming Solver

The OPTMODEL Procedure

Problem Summary
Objective Sense Maximization
Objective Function TotalProfit
Objective Type Linear
   
Number of Variables 30
Bounded Above 0
Bounded Below 0
Bounded Below and Above 30
Free 0
Fixed 0
Binary 30
Integer 0
   
Number of Constraints 56
Linear LE (<=) 56
Linear EQ (=) 0
Linear GE (>=) 0
Linear Range 0
   
Constraint Coefficients 112

Performance Information
Execution Mode Single-Machine
Number of Threads 1

Solution Summary
Solver MILP
Algorithm Branch and Cut
Objective Function TotalProfit
Solution Status Optimal
Objective Value 17500
   
Relative Gap 0
Absolute Gap 0
Primal Infeasibility 0
Bound Infeasibility 0
Integer Infeasibility 0
   
Best Bound 17500
Nodes 1
Iterations 10
Presolve Time 0.00
Solution Time 0.00

[1] Extract profit
1 1 0
2 1 0
3 1 0
4 0 -1500
5 1 0
6 1 1000
7 1 0
8 0 -1500
9 1 -1000
10 1 -1000
11 1 -1500
12 0 -2000
13 0 -1500
14 0 -1500
15 0 -2000
16 0 -2500
17 1 2000
18 1 2000
19 0 -2000
20 1 0
21 1 0
22 0 -4000
23 0 -2000
24 0 -2000
25 0 -5000
26 1 16000
27 0 4000
28 0 2000
29 0 0
30 0 2000