• During the rest of the year, this causes considerable costs, which increase for each week of storing. The unit cost for one time period of storing is assumed to be the cost corresponding
to about 2-3 weeks of storing. The cost for planning harvesting of an area with low accessibility is assumed to be in the same range, and this is an estimation of the risk taken
in that case. Some basic data and the size of the problem corresponding to the district used in the case study are summarized in Table 3. This district is typical for Holmen Skog as well
as other Swedish forest companies.
VII. Computational results
• The mathematical model is implemented in the language AMPL Fourer et al.
1993. The authors solve the MIP problem directly with the commercial software CPLEX 8.1. With a 1
tolerance from optimality, the problem is solvable directly. All computations are performed on a 2.66 GHz PC with 2 GB RAM memory.
A. Test problems
• The authors have generated five problems based on the basic case. The main purpose is to test the model and its practical usage. The second purpose is to test the performance of the
heuristic solution procedure and to give some input to a future, decision support system. This is an example of sensitivity analyses that Holmen Skog has defined are of interest.
• Problem 1, the basic case. Problem 2, accessibility at all areas and roads 100. The cost of harvesting an area or using a road with accessibility less than 100 is 0, which means road-
opening decisions are not considered. The road-opening variables are excluded and the constraints 9, 10, and 11 are relaxed.
• Problem 3, accessibility at all areas 100. The corresponding term in the objective is excluded.
• Problem 4, no storage cost because of quality deterioration. The corresponding two terms in the objective are excluded.
• Problem 5, no upper limit on harvesting capacity. The upper limit in constraints 12 is removed.
• Problem 6, simultaneous planning of two adjacent districts. Starting from Bergsjö case, an adjacent district is simulated and added to the basic case.
B. CPLEX directly
• The size of the six test problems and results using CPLEX directly are presented in Table 4. • CPLEX is the implementation of fundamental branch-and-bound technique and utilizes state-
of-the-art algorithms, including heuristics and a variety of branching and node selection strategies. Users can set duration limits, priorities, and tolerances on solution quality.
• In Table 4, Gap means the difference between best integer solution and best bound from LP relaxation divided with best bound, given as a percentage.
• Within 1 gap tolerance of optimality, we get solutions to all instances within a practical time limit. With default setting 0.01 gap, the solution time is more than 1 day. A solution
within 0.4 was found after 15 h, when that computation was interrupted. Coordination of two districts corresponds to a very large problem, and this is not solvable within a practical
time limit.
Universitas Sumatera Utara
C. Heuristic solution approach
