# Model of Multi-Branch Trees for Efficient Resource Allocation
## 2 Methods
### Procedure  
We explain the procedure of our method. First, we define all projects as objects for distributing a budget, departments, directors, and executives as defined above. The second paragraph of the Notation Subsection shows these terms with the example of Figure 1.  
**Step 1**: **Selection**  
First, all evaluators select projects that they must assess. However, all evaluators are prohibited from assessing them before all assessments are done by those who are children. In the example of Figure 1, \(E_{1}\) must wait until assessment by \(E_{3}\) and \(E_{4}\) is finished, and \(E_{0}\) must wait until all assessments by \(E_{1}\) to \(E_{8}\) are finished.  
A set of projects assessed by any evaluator is determined with the following procedure. If an evaluator is a director (who directly connects with leaf nodes), the director selects all projects in one's own department. In the example of Figure 1, a set of projects for \(E_{3}\) is \(\{P_{1},P_{2}\}\), and a set of projects for \(E_{5}\) is \(\{P_{5}\}\). If an evaluator is not a director (that is, either an executive or a mid-level executive), one selects a project among descendant leaves of one's child node each. In the example of Figure 1, \(E_{1}\) must select two projects; one is from descendant leaves of \(E_{3}\), and the other is from descendant leaves of \(E_{4}\). In this case, all combinations are \(\{P_{1},P_{3}\}\), \(\{P_{1},P_{4}\}\), \(\{P_{2},P_{3}\}\), and \(\{P_{2},P_{4}\}\). Among them, \(E_{1}\) selects a set. In the example of Figure 1, a set of projects for \(E_{1}\) is \(\{P_{1},P_{3}\}\). Note that the label of every branch in Figure 1 is a selected project.  
**Step 2**: **Evaluation**  
Second, all evaluators assess the projects that they must assess. They give a positive real number to each project as an assessment value. For capturing these values, we define a map \(v:\{0,\ldots,m\}\times\{1,\ldots,n\}\longrightarrow\mathbb{Z}\), where \(v(j,k)\) means the evaluation value that \(E_{j}\) gives \(P_{k}\).  
Note that an assessment is given relatively. Let us consider that evaluator \(E_{3}\) assesses two projects, \(P_{1}\) and \(P_{2}\), as shown in Figure 1. The assessment values of the two are denoted as \(v(3,1)\) and \(v(3,2)\), respectively. However, these values are relative to each other. That is, the case of \(E_{3}\) giving \(v(3,1)=2\) and \(v(3,2)=5\) is completely equivalent to the case of \(E_{3}\) giving \(v(3,1)=4\) and \(v(3,2)=10\).  
**Step 3**: **Calculation**  
In the final step, the final allocation of budgets is calculated. To do so, we first calculate an \(N\)-dimensional vector  
\[W(T_{(i,j)})=\Big{(}w(P_{1}),w(P_{2}),\ldots,w(P_{n})\Big{)}, \tag{1}\]  
which consists of rates of allocation. Therefore, \(\sum w(P_{i})=1\) (\(i\in\{1,\ldots,n\}\)). Using this vector, the executive distributes \(G\times w(P_{i})\) to \(P_{i}\), where \(G\) is the gross of the budget.  
Each element of \(W\) must correspond to each evaluator's assessment. That is, for any evaluator \(E_{j}\) and for any two projects \(P_{i}\) and \(P_{i^{\prime}}\) where \(E_{j}\) assesses both \(P_{i}\) and \(P_{i^{\prime}}\), \(v(j,i):v(j,i^{\prime})=w(P_{i}):w(P_{i^{\prime}})\). We developed a calculation method to solve \(W\) in the following subsections.