• The constraints 4 and 5 correspond to the storage balances at harvest areas and mills, respectively. The storage at terminals is directly included in the storage balance at mills. The
slack variable in eq. 5 is introduced to guarantee feasible solutions, and the penalty cost for slack assures that the demand will be satisfied if possible. The constraints 6 restrict the
total volume of purchased raw material, and constraints 7 and 8 specify the storage capacity at mills and terminals, respectively.
• Constraints 9 and 10 require at least one road connected to area i
to be open if transportation of logs from
i is done or if area i is harvested, respectively. These two
constraints are needed, as the logs can be transported from an area i in a later time period than the area was harvested. Constraints 11 specify the precedence relation between roads.
The set R
P
r is the set of roads connecting road r to the next level or higher level, if the
state road is the highest in the road hierarchy. Constraints 12 correspond to the restriction that the crews have a limited number of working days each month to use; the use of more
working days corresponds to a penalty cost. Finally, the variable restrictions are given in 13 and 14.
• The annual harvesting planning problem gives a large-scale, mixed-integer, linear problem. There are a number of binary variables corresponding to harvesting and road opening
decisions and continuous variables describing storage and flow. The linear relaxation of this model gives a good estimation of the objective value of the integer problem. For each
harvest area, the value of the corresponding binary harvest decision variable gives the proportion of the volume at that area, which is possible to use, to fulfill the demand,
constraints 3 and 4. The road network is highly aggregated, which gives that constraints 10 allow small values of
u
rt
. However, the cost corresponding to road opening is small compared with the total cost, so these constraints do not result in a weak lower bound from
the linear relaxation.
VI. A single district case study
• The case study comes from Bergsjö district in the Iggesund region. Holmen Skog is one of the largest forest owners in Sweden, with more than 10
6
hectares of forest. Holmen Skog is responsible for providing raw material to the Swedish wood-processing facilities included in
Holmen, consisting of a group of companies where saw-, pulp- and paper-mills are included. The Iggesund region is divided into six districts. The annual harvested volume at Bergsjö
district is about 250 000 m
3
. Holmen Skog owns the forest and also the mills. Annual harvest planning at a district aim to distribute the harvest areas during the year, ensuring that the
demand at the mills is satisfied and total cost is minimized. • The list of areas included in the case study corresponds to the pool of areas for the district.
At this time, the total supply of the areas included in the list corresponds to about 1.5 years of harvesting at the district. Areas in the case study are of various sizes, with total supplies
between 100 and 1000 m
3
and correspond to 1-20 days of work to harvest and forward. • There are five harvesting teams working full time in the district. The harvesting plan is made
for these harvesting crews working permanently at the district, with the possibility to utilize some overtime. The extra working days correspond to overtime or contracting of a part-time,
extra working team. One of the teams has two sets of machinery, and the number of possible extra working days corresponds to almost full-time schedule. In the model, it is
possible to utilize an unlimited number of extra days, above the normal overtime, to a penalty cost. There are five mills in the case study with different production. Two of the mills
can use a terminal for storage.
• The total planning horizon in this case study is 12 months. All decisions considering harvesting, storage, and transportation are discretely made at a monthly level. Storage cost
at harvest areas corresponds to decrease of value because of quality deterioration. The decrease of quality is negligible during the frozen part of the year. In the north of Sweden,
this usually corresponds to December, January, and February.
Universitas Sumatera Utara
• 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
