## 2 Materials and Methods
### Our Optimization Method/Algorithm by Tree Search  
Motivated by the principles of MCTS, we will introduce a simple tree search method/algorithm to find the \"_optimal_\" arrangement, with the _largest_ long-term _Waste Heat Recovery Ratio_ (%):  
\[E^{\prime}=\frac{M_{ar}\cdot(C_{ar,\rho}\cdot T_{ar,\rho}-C_{ar,i}\cdot T_{ar,i })}{M_{ex}\cdot C_{ex,i}\cdot T_{ex,j}}\cdot 100,\text{definedin }(30),\]  
of heat-storage ceramic bricks (checkers) in a heat regenerator, when there are \(p\) different types of checkers that can be selected for _each_ of the \(n\) positions in a fixed-bed heat regenerator. In our following discussions, a _node_ of search tree simply represents an _arrangement_ of checkers at \(n\) positions in a fixed-bed heat regenerator. We begin with the notion of a \"_Partition_\" of the set  
\[S=\{1,\cdots,n\}\]  
of positions, to be placed by checkers, in a fixed-bed heat regenerator. A _Partition_\(\alpha\) of \(S\) is a subset of \(S=\{1,\cdots,n\}\) such that the largest integer \(n\) must be included in \(\alpha\). **Example 1. ** _For the set \(S=\{1,\cdots,9\}\), \(\alpha=\{4,9\}\) and \(\beta=\{4,6,9\}\) are Partitions of \(S\). Assume that_  
\[\alpha=\{c_{1},\cdots,c_{m}\}\text{ with }c_{1}<\cdots<c_{m}=n \tag{1}\]  
_is a Partition of \(S=\{1,\cdots,n\}\). Using the positive integers \(c_{1},\cdots,c_{m}=n\) in \(\alpha\), we may create the following decomposition_  
\[S=I_{1}\cup\cdots\cup I_{m} \tag{2}\]  
_for \(S=\{1,\cdots,n\}\), in which_  
\[I_{1}=\{1,\cdots,c_{1}\},\ I_{2}=\{c_{1}+1,\cdots,c_{2}\},\cdots,\ I_{m}=\{c_ {m-1}+1,\cdots,c_{m}=n\}. \tag{3}\]  
**Example 2. ** _For the Partition \(\alpha=\{4,9\}\) of the set \(S=\{1,\cdots,9\}\), we have the following decomposition_  
\[S=I_{1}\cup I_{2}\text{ with }I_{1}=\{1,2,3,4\}\text{ and }I_{2}=\{5,6,7,8,9\}\]  
_For the Partition_  
\[\alpha=\{c_{1},\cdots,c_{m}\}\text{ with }c_{1}<\cdots<c_{m}=n\]of the \(S=\{1,\cdots,n\}\), we will conveniently say that each \(I_{k}\) is an \"Interval Component\" of \(S\) with respect to \(\alpha\), because each \(I_{k}\) contains certain consecutive positive integers taken from \(S=\{1,\cdots,n\}\). This phenomenon is clear from the example (Example 2) shown above._  
We have the following simple facts about the decomposition (2) of \(S=\{1,\cdots,n\}\). **Fact A. **_The endpoints of the Interval Components \(I_{k}\) in (2) simply constitute the Partition set \(\alpha\) of \(S\)._  
**Fact B. **_The Interval Components of the decomposition (2) of \(S=\{1,\cdots,n\}\), with respect to the Partition \(\alpha\), are disjoint._  
**Fact C.**_Elements of the Interval Components of the decomposition (2) of \(S=\{1,\cdots,n\}\) are linearly ordered._  
There is a _Partial Ordering_ on the collection of all _Partitions_ of \(S=\{1,\cdots,n\}\). For Partitions \(\alpha\) and \(\beta\) of \(S=\{1,\cdots,n\}\), we say that \(\beta\) is _finer_ than \(\alpha\) if and only if  
\[\alpha\subset\beta\]  
We say that \(\beta\) is _strictly finer_ than \(\alpha\) if and only if  
\[\alpha\subset\beta\text{ and }\beta\
eq\alpha.\]  
We will use the notation  
\[\alpha\prec\beta\]  
to indicate the condition that \(\beta\) is _strictly finer_ than \(\alpha\). **Example 3. **_For Partitions \(\alpha=\{4,9\}\) and \(\beta=\{4,6,9\}\) of \(S=\{1,\cdots,9\}\), we have \(\alpha\prec\beta\)._  
Now we discuss the notion of \"_qualified_ arrangements with respect to a _Partition_\". Assume that \(\alpha\) is a _Partition_ of \(S=\{1,\cdots,n\}\) as shown in (1). We say that an _arrangement_\(\mathbf{U}\) of checkers on \(S=\{1,\cdots,n\}\) is a \"_qualified_ arrangement with respect to the _Partition_\(\alpha\)\" if and only if the following condition (4) is satisfied. **Definition 1. **_We say that an arrangement \(\mathbf{U}\) of checkers on \(S=\{1,\cdots,n\}\) is a \"qualified arrangement with respect to the Partition \(\alpha\)\" if the following condition is satisfied.