1

Fig. 1. The map of Portugal and the study area.

2

Fig. 2. Maps of study case area presenting harvest schedule generated by Simulated Annealing (SA) for problem III. Stands in black are harvested stands in the corresponding period. Problem III corresponded to the maximization of net returns using even-flow of harvest constraints, ending volume inventory and carbon stock level constraints and adjacency constraints.

3

Fig. 3. Performance of the different implementations of Simulated Annealing (SA) (one, two and three-opt moves) for all problem formulations and for the instance of 1000 stands for different combinations of initial temperature (2, 7, 12, 100, 100000) and cooling schedule (0.8, 0.99996). The graph shows the best, the average and the worst solution for each implementation problem. Problem I encompassed the use of even-flow of harvest constraints, Problem II further included the use of ending volume inventory and carbon stock level constraints. Problem III was an extension of previous models and further included adjacency constraints.

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Fig. 4. Comparison of cooling scheduling (0.8 and 0.99996) and initial temperature (2, 7, 12, 100, 100000) for each problem formulation for the instance 1000 stands. Problem I encompassed the use of even-flow of harvest constraints, Problem II further included the use of ending volume inventory and carbon stock level constraints. Problem III was an extension of previous models and further included adjacency constraints.

Table 1. Configuration of parameters (initial temperature T0, cooling schedule ζ ) for the Instance of 1000 stands. Problem I encompassed the use of even-flow of harvest constraints, Problem II further included the use of ending volume inventory and carbon stock level constraints. Problem III was an extension of previous models and further included adjacency constraints.
Model Average objective Best
objective 		Worst
objective 		Average time[s] GAP1 1
[%] 		GAP2 2
[%] 		Best
parameters
Problem I 253654092.6 254242073 252974250 11560 0.2313 0.499 T0 = 100000,
ζ = 0.8
Problem II 254044417.2 254553811 253337282 13913 0.2 0.478 T0 = 100000,
ζ = 0.99996
Problem III 253998184.9 254495562 253219220 20134 0.1954 0.502 T0 = 100000,
ζ = 0.99996
1 GAP1 refers to the relative gap between the best and the average value.
2 GAP2 refers to the relative gap between the best objective value found and the worst value achieved by SA.
5

Fig. 5. Performance of the different implementations of Simulated Annealing (SA) (one, two and three-opt moves) for problem formulation III and for the instance of 1000 stands for different combinations of initial temperature and cooling schedule. Problem III corresponded to the maximization of net returns using even-flow of harvest constraints, ending volume inventory and carbon stock level constraints and adjacency constraints.

Table 2. Comparison of performance of exact (CPLEX) and heuristic (Simulated Annealing (SA)) methods.
Stands Problem CPLEX SA Statistics
    Best objective Time[s] Gap[%]3 Best objective Average time[s] Saved time[%] Relative gap[%]4
100 I 19385252 6591 3.78 18208598 784 88 6.07
100 II 18787533 24348 6.93 18247907 1039 96 2.87
100 III 19987548 32813 0.51 18244260 1523 95 8.72
250 I 49576379 80 0.85 47641286 2248 –2710 3.90
250 II 49705423 148 0.58 47762817 2775 –1775 3.91
250 III 49535258 10890 0.92 47787089 4212 61 3.53
500 I 110140532 405 0.14 107751035 4967 –1126 2.17
500 II 109921875 523 0.33 107789228 6416 –1127 1.94
500 III 110059776 13460 0.20 107729193 9448 30 2.12
1000 I 261614671 8459 0.01 254242073 11560 –37 2.82
1000 II 261604515 34076 0.01 254553811 13913 59 2.70
1000 III - - - 254495562 20134 n.a. n.a.
3 The column “Gap” refers to the default gap calculation in CPLEX and stands for the relative difference between the highest objective value and the lowest upper bound that is found during optimization process [%].
4 The column “Relative gap” refers to the relative percentage difference between the CPLEX and SA solutions.