## Materials and Methods
### Supply-Demand Interaction: Development of Competition Scenarios  
There is often more than one industry operating simultaneously in a given area. When a particular land unit U\({}_{\text{ij}}\) is harvested by one industry it will become unavailable to others. In our model we used optimization (Integer Linear Programming) to define which competitor harvests which land unit. Since we worked with 1-ha cells (U\({}_{\text{ij}}\)) that cannot be subdivided, variables are of type integer. Considering that finding a solution for problems of this type is mathematically and computationally demanding, we used a Branch and Bound (B&B) algorithm [61] to obtain an approximately optimal solution, within a defined threshold. We addressed the biomass supply-demand relationship based on the following steps:  
1. **Maximizing value**: maximization of the value (maximum resource suitability) for each spatial unit; it was assumed that all industries seek the highest resource value and are willing to pay the price corresponding to its value. Due to the large number of spatial units in the study area, we established an automatic routine in C# to formalize the problem following the Integer Linear Programming (ILP) scheme in Equation (1) \[Max\Big{(}\sum(f_{value})\Big{)}\] (1) subject to \[\forall T_{i}\sum\left(quantity\cdot Element(i,j)\right)\in T_{i}\leq Max( Vol_{T_{i}})\cap\geq Min(Vol_{T_{i}})\] (2) \[\forall Element(i,j)\subset(0,1)\] (3) Counters \[Sup=\sum_{i}Element(i,j)\] (4)  
Figure 6: Ranking of weights for criteria and sub-criteria for t\({}_{0}\) (blue bars) used in the generic value model and trends over a period of 20 years (t\({}_{20}\), orange bars) in the Nordeste region, Portugal. \[Vol=\sum\left(quantity\cdot Element(i,j)\right)\] (5) Variable type \[Int\
u Element(i,j)\in T_{i} \tag{6}\]
2. **Solving** the problem: solving the ILP scheme in (1) using an approximation of optimal solution by the B&B method applied with the open source tool lp_solve 5.5 IDE [62]; B&B was restricted to a threshold (GAP) of 10% and stopped when the first solution was found;
3. **Writing the solution**: production of a new forest spatial distribution considering changes in each U\({}_{\rm ij}\) in time when a solution in (2) was found. We applied the following rules to the supply-demand interaction model: (i) The value of the resource is established based on its demand through a set of criteria representing the valuation of demand according to each industry; (ii) The price of the resource is related to its value (maximum value, maximum price; minimum value, minimum price; (iii) Forests can be harvested when three conditions are simultaneously met: 1. Availability--the resource is available when it can be exploited according to economic, social and environmental conditions and local national legal frameworks; 2. Price--the price of the resource satisfies the two parts involved in the process: the buyer and the seller; 3. Quantity--there is enough resource for the sustainable operation of an industry;
4. Forest resources can only be extracted (through thinning or harvesting) by one of the competing industries. When a land unit meets all the rules above, it will be used (harvested or thinned) by the industry that pays the highest price. To evaluate the supply-demand interaction regarding objective 5 we used the following indicator: **Id 7**: Cumulative managed area (ha)--cumulative forest area where thinning was applied over time. Moreover, the interaction between supply and demand among industries can increase competition, which should be minimized according to objective 6, assessed based on indicator Id 8: **Id 8**: Volume not harvested due to competition (m\({}^{3}\))--calculated as the summation of the differences between the industry harvests and the minimum value between the maximum harvest volume goal and the potential volume available for harvesting for each industry.