text:
\section{Introduction}
\label{Introduction}

Analytic hierarchy process (AHP) is a multi-criteria decision-making method that helps the decision-makers facing a complex problem with multiple conflicting and subjective criteria\cite{ishizaka2009analytic}.
AHP has been widely applied in various areas\cite{zhu2016hesitant,lin2015measuring,yepes2015cognitive,wang2011linear}.
Based on a hierarchical structure, the criteria weights can be derived from the pairwise comparison matrix.
Pairwise comparison matrix as a crucial component of AHP, is commonly utilized to estimate preference values of finite alternatives with respect to a given criterion.

However, the pairwise comparison matrix is difficult to complete in the following cases, which
obstructs the subsequent operations of the classical AHP.

\begin{itemize}
\item The experts lack the knowledge of one or more alternatives.
\item Partial data in pairwise comparison matrix has been lost.
\item Huge number of pairwise comparison requested: $(n^2-n)/2$ for n alternatives. For example, 8 alternatives and 6 criteria require 183 entries\cite{ishizaka2011review}.
\end{itemize}

Hitherto, it has attracted widespread attention over the issue of incomplete pairwise comparison matrix in AHP.
There are quantities of methods having been developed to address the incomplete comparison matrices.
Some of them use the incomplete matrix to assign priority weight to each alternative without completion\cite{harker1987incomplete,carmone1997monte,hua2008ds,xu2013logarithmic,dopazo2011parametric}. Some methods are devoted to estimate the missing values in matrix\cite{fedrizzi2007incomplete,ergu2016estimating,bozoki2010optimal,chen2015bridging,zhang2012linear,hu2006backpropagation,alonso2008consistency,benitez2015consistent,gomez2010estimation,buyukozkan2012new,cabrerizo2010managing}.
While these methods have a good effect on allocating the priority weights to alternatives, the methods that focus on matrix completion are more ``well-rounded'' to restore the preference relations between each pair of alternatives, which are inaccessible before completion.

After reviewing the existing completion methods for incomplete pairwise comparison matrix in AHP, it can be found that the vast majority of these methods focus on the optimal consistency of pairwise comparison matrix\cite{fedrizzi2007incomplete,ergu2016estimating,bozoki2010optimal,chen2015bridging,zhang2012linear}.
For example,
Fedrizzi and Giove\cite{fedrizzi2007incomplete} proposed a new method to complete the pairwise comparison matrix by minimizing its global inconsistency.
Ergu \emph{et al.}\cite{ergu2016estimating} extended the geometric mean induced bias matrix to estimate the missing judgments and improve the consistency ratios by the least absolute error method and the least square method.
Based on the graph theory, Bozo\'iki \emph{et al.}\cite{bozoki2010optimal} developed a completion method which focuses on the optimal consistency and estimates the missing values through iteration.
Besides,
Hu and Tsai\cite{hu2006backpropagation} proposed a well-known back propagation multi-layer perception to estimate the missing comparisons of incomplete pairwise comparison matrices in AHP.
Alonso \emph{et al.}\cite{alonso2008consistency} defined a way to calculate the consistency of incomplete pairwise comparison matrix and maintain its consistency level after completion.
Ben\'itez \emph{et al.}\cite{benitez2015consistent} used the Frobenius norm to obtain the most consistent/similar complete matrix with the initial incomplete matrix.
Gomez \emph{et al.}\cite{gomez2010estimation} proposed a completion method based on the neural network. These methods are useful but have some limitations:

\begin{itemize}
\item Most methods complete the pairwise comparison matrix with optimal consistency, which sometimes can not restore the incomplete matrix to its original state, especially when the initial matrix is not high consistent.
\item The methods are computationally expensive.
\end{itemize}

Taking into account these issues, a DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP is proposed.
Based on DEMATEL method which has ability to derive the total relation matrix from direct relation matrix, the completion of incomplete pairwise comparison matrix can be divided into three steps.

\begin{itemize}
\item[-] Firstly, turn the incomplete comparison matrix into direct relation matrix.
\item[-] Secondly, DEMATEL method is adopted to obtain the total relation matrix.
\item[-] Thirdly, transform the total relation matrix into pairwise comparison matrix with reciprocal preference relations.
\end{itemize}

The proposed method provides a new perspective to estimate the missing values in pairwise comparison matrix with explicit physical meaning (i.e. by calculating the total relations among alternatives). Besides, the proposed method has low computational cost.

The rest of this paper is organized as follows. Section 2 introduces some preliminaries of this work. Section 3 illustrates the procedure of the proposed method and an application for ranking tennis players is presented. Section 4 discusses the superiority and efficiency of the proposed method and section 5 ends the paper with the conclusion.

\section{Preliminaries}
\label{Rreparatorywork}

\subsection{Analytic Hierarchy Process (AHP)}

AHP is a multi-criteria decision-making method developed by Saaty\cite{saaty1980analytic} and aims at quantifying relative weights for a given set of criteria on a ratio scale\cite{xu2015deviation}.
As a decision-making approach, it permits a hierarchical structure of criteria, which provides users with a better focus on specific criteria and sub-criteria when allocating the weights\cite{ishizaka2013multi}.
Besides, AHP has ability to judge the consistency of pairwise comparison matrix to show its potential conflicts in decision-making process.
AHP has been widely used in
supply chain management\cite{chan2007global,deng2014supplier,chan2006ahp},
human reliability analysis\cite{su2015dependence},
cloud computing\cite{liu2016decision},
energy planning\cite{ishizaka2016energy,onar2015multi,kahraman2009comparative},
site selection\cite{erdougan2016combined,lee2015integrated,pourahmad2015combination},
software assessment\cite{hu2013software,hu2015cost} and
performance evaluation\cite{zavadskas2015selecting,wu2010evaluating}.
AHP can be extended further by other theories and methods such as fuzzy number\cite{chan2008global,wang2008extent}, choquet integral\cite{corrente2015combining}, k-means algorithm\cite{lolli2014new}, VIKOR\cite{buyukozkan2015evaluation} and TOPSIS\cite{erdogan2014evaluating,kahraman2015fuzzy}.

Generally, the process of applying AHP is divided into three steps.
Firstly, establish a hierarchical structure by recursively decomposing the decision problem.
Secondly, construct the pairwise comparison matrix to indicate the relative importance of alternatives.
A numerical rating including nine rank scales is suggested, as shown in Tab.\ref{Rating}.
Thirdly, verify the consistency of pairwise comparison matrix and calculate the priority weights of alternatives. For the completeness of explanation, several basic concepts and formulas are introduced as follows.

\begin{table}[H]
\centering
\caption{Numerical rating in AHP}
\label{Rating}
 \footnotesize

\begin{tabular*}{0.65\textwidth}{@{\extracolsep{\fill}}ll}
 \toprule[1pt]
\quad Scale & Meaning \\ \hline
\quad 1 & Equal importance \\
\quad 3 & Moderate importance \\
\quad 5 & Strong importance \\
\quad 7 & Demonstrated importance \\
\quad 9 & Extreme importance \\
\quad 2, 4, 6, 8 & Intermediate values \\
\bottomrule[1pt]
\end{tabular*}
\end{table}

\begin{myDef}
(Pairwise comparison matrix)
Assume $\begin{Bmatrix} E_1, E_2, \cdots, E_n \end{Bmatrix}$ are $n$ alternatives available for decision-making, the pairwise comparison matrix is indicated as $M=\left ( m_{ij} \right )_{n \times n} \left (i,j=1,2,\cdots ,n\right )$, and satisfies:
\begin{equation}
\label{PCM}
m_{ij}=\left\{\begin{matrix}
\frac{1}{m{ji}} & i\neq j\\
1 & i=j
\end{matrix}\right.
\end{equation}
where $m_{ij}$ represents the relative importance of $E_{i}$ over $E_{j}$.
\end{myDef}

Consistency checking is introduced in AHP to verify the usability of constructed pairwise comparison matrix.

\begin{myDef}
(Consistency ratio)
For the pairwise comparison matrix $M_{n \times n}$,
let $\lambda _{\max }$ denote the largest eigenvalue of $M$, consistency index ($CI$) is defined as
\begin{equation}
\label{CI}
CI = \frac{{\lambda _{\max }  - n}}{{n - 1}}
\end{equation}
Based on $CI$, consistency ratio (CR) is defined as

\begin{equation}
\label{CR}
CR = \frac{{CI}}{{RI}}
\end{equation}
where $RI$ is the random consistency index related to the dimension of matrix, listed in Tab.\ref{RI}.

\begin{table}[H]
\centering
\caption{Random consistency index $RI$}
\label{RI}
 \footnotesize

\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}cccccccccccc}
\toprule[1pt]
 \multicolumn{2}{c}{n} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9  &10  \\ \hline
\multicolumn{2}{c}{$RI$} & 0 & 0 & 0.52 & 0.89 & 1.12 & 1.26 & 1.36 & 1.41 & 1.46 & 1.49  \\
\bottomrule[1pt]
\end{tabular*}
\end{table}

\end{myDef}

If $CR<0.1$, the constructed pairwise comparison matrix is considered acceptable and the priority weights of alternatives can be obtained through the eigenvector as Def.\ref{eigenvector}.
Otherwise, the comparison matrix needs to be reconstructed.

\begin{myDef}
\label{eigenvector}
(Eigenvector)
For the pairwise comparison matrix $M_{n \times n}$ with acceptable consistency,
suppose $\vec{w} = ( {w_1 ,w_2 ,\cdots,w_n })^T $ is the eigenvector of $M$, whose $w_i(i=1, 2, \cdots, n)$  is indicated as the priority weight of the $i$-th alternative and calculated by
\begin{equation}
\label{weights of AHP}
M\vec{w} = \lambda _{\max }\vec{w}
\end{equation}
\end{myDef}

\subsection{DEMATEL method}
Decision-making trial and evaluation laboratory (DEMATEL) method was originally developed by Battelle Memorial Institute of Geneva Research Center, aimed at the fragmented and antagonistic phenomena of world societies and searched for integrated solution\cite{gabus1972world,fontela1976dematel}.
DEMATEL is built on the basis of graph theory,
effectively assessing the total relations and constructing a map between system components in respect to their type and severity\cite{tzeng2007evaluating}.
Reviewing the former studies, DEMATEL has been successfully applied in many diverse areas such as
service quality analysis\cite{shieh2010dematel,tseng2009causal},
stock selection\cite{shen2015combined} and
supply chain management\cite{ ISI:000368039100008,wu2015case,wu2015exploring}.
DEMATEL can be further extended by other theories and methods such as fuzzy numbers\cite{mentes2015fsa,tsai2015using} and ANP\cite{ ISI:000356990800033,tzeng2012combined, ISI:000262178000049,wu2008choosing}.

The procedure of DEMATEL is divided into five steps.

\emph{\textbf{Step 1:} Define quality feature and establish measurement scale}

Quality feature is a set of influential characteristics that impact the sophisticated system, which can be determined by literature review, brain-storming, or expert evaluation.
After defining the influential characteristics in researching system, establish the measurement scale for the causal relationship and pairwise comparison among influential characteristics.
Four levels 0,1,2,3 are suggested, respectively meaning ``no impact'', ``low impact'', ``high impact'' and ``extreme high impact''. In this step, factors and their direct relations are displayed by a weighted and directed graph.

\emph{\textbf{Step 2:} Extract the direct relation matrix of influential factors}

In this step, transformation from the weighted and directed graph into direct relation matrix has been done.
For $n$ influential characteristics $F_1, F_2, \cdots, F_n$, direct relation matrix is denoted as $D=\left(d_{ij}\right)_{n \times n} \left (i,j=1,2,\cdots ,n\right )$,
where $d_{ij}$ is the direct relation of $F_i$ over $F_j$ based on the measurement scale and satisfies $d_{ij}=0$ if $i=j$.

\emph{\textbf{Step 3:} Normalize the direct relation matrix}

The normalized direct relations of factors are a mapping from $d_{ij}$ to $\left [ 0,1 \right ]$, defined as below.

\begin{myDef}
\label{NorDRM}
(Normalized direct relation matrix)
For the framework of $n$ influential characteristics $\begin{Bmatrix} F_1, F_2, \cdots, F_n \end{Bmatrix}$, normalized matrix $N$ of direct relation matrix $D=\left(d_{ij}\right)_{n \times n} \left (i,j=1,2, \cdots ,n\right )$ is obtained by
\begin{equation}
\label{normMatrix}
N = \frac{D}{\max (\mathop { } \sum\limits_{j = 1}^n {{d_{ij}},} \mathop {} \sum\limits_{i = 1}^n {{d_{ij}}} )}
\end{equation}
\end{myDef}

Note that there must be at least one row of matrix $D$ satisfying $\sum_{j=1}^{n}d_{ij}<{\max (\mathop { } \sum\limits_{j = 1}^n {{d_{ij}},}\mathop {} \sum\limits_{i = 1}^n {{d_{ij}}} )} $\cite{lin2008causal}. Hence,
based on the studies of \cite{papoulis1985probability} and \cite{goodman1988introduction}, sub-stochastic matrix can be obtained through utilizing the normalized direct relation matrix $N$
and
absorbing state of Markov chain matrices, shown in (\ref{normMatrix2}).
\begin{equation}
\label{normMatrix2}
\mathop {\lim }\limits_{k \to \infty } N^k=O \quad and \quad \mathop {\lim }\limits_{k \to \infty } (I + N + N^2  +  \cdots  +N^k ) = N(I - N)^{ - 1}
\end{equation}
where $O$ is the null matrix and $I$ is the identity matrix.

\emph{\textbf{Step 4:} Calculate the total relation matrix}

Due to the characteristic of normalized direct relation matrix $N$, total relation matrix which contains direct and indirect relations among factors can be derived from (\ref{TRM}).

\begin{myDef}
(Total relation matrix)
For the framework of $n$ influential characteristics $\begin{Bmatrix} F_1, F_2, \cdots, F_n \end{Bmatrix}$, assume $N=\left(n_{ij}\right)_{n \times n} \left (i,j=1,2,\cdots ,n\right )$ is the normalized direct relation matrix, total relation matrix $T$ is defined as
\begin{equation}
\label{TRM}
T =\mathop {\lim }\limits_{k \to \infty } (N + N^2  +  \cdots  +N^k )= N(I - N)^{ - 1}
\end{equation}
where $I$ is the identity matrix.
\end{myDef}

\emph{\textbf{Step 5:} Construct the DEMATEL map}

Based on the sum of each row $R_i(i=1,2,\cdots,n)$ and column $C_i$ $(i=1,2,\cdots,n)$ of the total relation matrix $T_{n \times n}$, $(R_i+C_i)$ and $(R_i-C_i)$
can be obtained.
$(R_i+C_i)$ is defined as the prominence, indicating the importance of the i-th influential factor.
$(R_i-C_i)$ classifies the i-th influential factor into the cause or effect category in researching system. If $(R_i-C_i)>0$, the factor is regarded as a cause factor.
If $(R_i-C_i)<0$, the factor is an effect one. The DEMATEL map is shown as Fig.\ref{DEMATELdiagram}.

\begin{figure}[H]
\begin{center}
\psfig{file=DEMATELdiagram.eps,scale=0.56,angle=0}
\caption{DEMATEL map}
\label{DEMATELdiagram}
\end{center}
\end{figure}

\begin{figure}[H]
\begin{center}
\psfig{file=methodFlowchart.eps,scale=0.51,angle=0}
\caption{Procedure of DEMATEL-based completion method}
\label{flowchart}
\end{center}
\end{figure}

\newpage
\section{DEMATEL-based completion method}
\label{ProposedMethod}
In this section, a DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP is proposed. Moreover, an application for ranking top tennis players is presented.

\subsection{Procedure of DEMATEL-based completion method}

In this sub-section, the procedure of DEMATEL-based completion method is presented and illustrated by an example. Fig.\ref{flowchart} shows the process of the proposed method.

Assume an $4\times4$ incomplete pairwise comparison matrix $M$ and the unavailable/missing values are displayed by `*'.

\begin{displaymath}
M = \left[ {\begin{array}{*{20}c}
1 & * & 4 & 8\\
* & 1 & 2 & 4\\
1/4 & 1/2 & 1 & 2 \\
1/8 & 1/4 & 1/2 & 1
\end{array}}\right]
\end{displaymath}

\emph{\textbf{Step 1:} From incomplete pairwise comparison matrix to direct relation matrix.}

Pairwise comparison matrix reflects the preference relations between each pair of factors,
where $m_{ij}(i,j=1,2,3,4)$ in $M$ indicates the relative importance of factor $i$ to factor $j$. Hence, the known values in $M$ can be put into the direct relation matrix $D$ without any transformation and the unavailable ones(`*') are substituted by 0. The direct relation matrix $D$ is as:

\begin{displaymath}
D = \left[ {\begin{array}{*{30}c}
1 & 0 & 4 & 8\\
0 & 1 & 2 & 4\\
1/4 & 1/2 & 1 & 2 \\
1/8 & 1/4 & 1/2 & 1
\end{array}}\right]
\end{displaymath}

\emph{\textbf{Step 2:} From direct relation matrix to total relation matrix. }

First of all, normalize the direct relation matrix $D$ according to (\ref{normMatrix}). Then, based on the normalized matrix $N$ and (\ref{TRM}), total relation matrix $T$ can be obtained. The normalized direct relation matrix $N$ and total relation matrix $T$ are calculated as:
\begin{displaymath}
N = \left[ {\begin{array}{*{30}c}
0.0667 & 	0.0000 &	0.2667 &	0.5333 \\
0.0000 &	0.0667 &	0.1333 &	0.2667 \\
0.0167 &	0.0333 &	0.0667 &	0.1333 \\
0.0083 &	0.0167 &	0.0333 &	0.0667
\end{array}}\right]
,
T = \left[ {\begin{array}{*{30}c}
0.0835 &	0.0241 &	0.3371 &	0.6742 \\
0.0060 &	0.0835 &	0.1685 &	0.3371 \\
0.0211 &	0.0421 &	0.0899 &	0.1798 \\
0.0105 &	0.0211 &	0.0449 &	0.0899
\end{array}}\right]
\end{displaymath}

\emph{\textbf{Step 3:} From total relation matrix to pairwise comparison matrix.}

In fact, the total relation matrix has completed the unavailable values in initial incomplete matrix.
Nevertheless, in pairwise comparison matrix, the multiplication of each pair of symmetric values along the diagonal line is required to equal 1, yet the values in total relation matrix are between 0 and 1.
Based on (\ref{transTotal2PCM}), it is feasible to accomplish this transformation from the total relation matrix $T=(t_{ij})_{n\times n}(i,j=1,2,\cdots,n)$ to pairwise comparison matrix $M_c=(c_{ij})_{n\times n} (i,j=1,2,\cdots,n)$.

\begin{equation}
\label{transTotal2PCM}
\left\{ {\begin{array}{*{20}l}
\frac{t_{ij}}{c_{ij}}=\frac{t_{ji}}{c_{ji}}  \\
c_{ij}=\frac{1}{c_{ji}}
\end{array}} \right.
\end{equation}

Hence, the pairwise comparison matrix transformed from $T_{4 \times 4}$ is:
\begin{displaymath}
M_c = \left[ {\begin{array}{*{30}c}
1.0000 &	2.0000 &	4.0000 &	8.0000 \\
0.5000 &	1.0000 &	2.0000 &	4.0000 \\
0.2500 &	0.5000 &	1.0000 &	2.0000 \\
0.1250 &	0.2500 &	0.5000 &	1.0000
\end{array}}\right]
\end{displaymath}

Finally, based on the proposed method, the completion for incomplete pairwise comparison matrix $M$ has been accomplished as:

\begin{displaymath}
M = \quad
\left[ {\begin{array}{*{20}c}
1 & * & 4 & 8\\
* & 1 & 2 & 4\\
1/4 & 1/2 & 1 & 2 \\
1/8 & 1/4 & 1/2 & 1
\end{array}}\right]
\rightarrow
\left[ {\begin{array}{*{20}c}
1 & \textbf{2} & 4 & 8 \\
\textbf{1/2} & 1 & 2 & 4\\
1/4 & 1/2 & 1 & 2 \\
1/8 & 1/4 & 1/2 & 1
\end{array}}\right]
\end{displaymath}

\subsection{An application for ranking tennis players}

Professional tennis as a global popular sport owns large numbers of viewers around the world.
The professional tennis associations such as ATP have been collecting data about the tournaments and the players.
There is a free available database that collects the results of the top tennis players since 1973 (see at http://www.atpw-\\orldtour.com/Players/Head-To-Head.aspx).

In original statistics, 25 tennis players who have been the champion on ATP since 1973 are selected. The statistical competition results matrix is shown in Tab.\ref{statisticalMatrix}.
However, the statistical matrix is not complete since some of players did not have the chance to compete with each other or the partial data has been lost. Thus, DEMATEL-based completion method is adopted to rank the tennis players based on these statistics. The process is divided into three steps, which is shown in Fig.\ref{rankFlowchart}.

\begin{figure}[htbp]
\begin{center}
\psfig{file=rankingFlowchart.eps,scale=0.16,angle=0}
\caption{Flowchart for ranking tennis players}
\label{rankFlowchart}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\psfig{file=transformation.eps,scale=0.68,angle=0}
\caption{Rule of transformation}
\label{transformation}
\end{center}
\end{figure}

\newgeometry{left=3cm,bottom=0.4cm,top=0.4cm}
\begin{sidewaystable}
\centering
\caption{Statistical competition results matrix: the wins/total number of matches}
\label{statisticalMatrix}

 \tiny
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}llllllllllllllllllllllllll}
\toprule[1pt]
           & Aga & Bec & Bor & Con & Cou & Djo & Edb & Fed & Fer & Hew & Kaf & Kue & Len & McE & Moy & Mus & Nad & Nas & New & Raf & Rio & Rod & Saf & Sam & Wil \\ \hline
Agassi     &        & 10/14  &      & 2/2     & 5/12    &          & 6/9    & 3/11    & 2/5     & 4/8    & 8/12       & 7/11    & 2/8   & 2/4     & 3/4  & 5/9    & 0/2   &         &          & 10/15  & 1/3  & 5/6     & 3/6   & 14/34   & 5/7      \\
Becker     & 4/14   &        &      & 6/6     & 6/7     &          & 25/35  &         &         & 1/1    & 4/6        &         & 10/21 & 8/10    & 2/4  & 2/3    &       & 1/1     &          & 2/3    & 3/5  &         & 0/1   & 7/19    & 7/10     \\
Borg       &        &        &      & 15/23   &         &          &        &         &         &        &            &         & 6/8   & 7/14    &      &        &       & 10/15   & 1/4      &        &      &         &       &         & 1/1      \\
Connors    & 0/2    & 0/6    & 8/23 &         & 0/3     &          & 6/12   &         &         &        &            &         & 13/34 & 14/34   &      &        &       & 12/27   & 2/4      &        &      &         &       & 0/2     & 0/5      \\
Courier    & 7/12   & 1/7    &      & 3/3     &         &          & 6/10   &         &         &        & 1/6        & 1/1     & 0/4   & 2/3     & 2/3  & 7/12   &       & 0/1     &          & 0/3    & 0/3  &         & 1/2   & 4/20    &          \\
Djokovic   &        &        &      &         &         &          &        & 15/31   & 2/3     & 6/7    &            &         &       &         & 2/4  &        & 17/39 &         &          &        &      & 4/9     & 0/2   &         &          \\
Edberg     & 3/9    & 10/35  &      & 6/12    & 4/10    &          &        &         &         &        & 1/3        &         & 14/27 & 6/13    & 1/1  & 10/10  &       &         &          & 3/3    & 1/1  &         &       & 6/14    & 9/20     \\
Federer    & 8/11   &        &      &         &         & 16/31    &        &         & 10/13   & 18/26  & 2/6        & 1/3     &       &         & 7/7  &        & 10/32 &         &          & 0/3    & 2/2  & 21/24   & 10/12 & 1/1     &          \\
Ferrero    & 3/5    &        &      &         &         & 1/3      &        & 3/13    &         & 4/10   & 1/3        & 3/5     &       &         & 8/14 &        & 2/9   &         &          & 2/3    & 3/4  & 0/5     & 6/12  &         &          \\
Hewitt     & 4/8    & 0/1    &      &         &         & 1/7      &        & 8/26    & 6/10    &        & 7/8        & 3/4     &       &         & 7/12 &        & 4/10  &         &          & 3/4    & 3/5  & 7/14    & 7/14  & 5/9     &          \\
Kafelnikov & 4/12   & 2/6    &      &         & 5/6     &          & 2/3    & 4/6     & 2/3     & 1/8    &            & 5/12    &       &         & 3/6  & 1/5    &       &         &          & 3/5    & 6/8  &         & 2/4   & 2/13    & 1/2      \\
Kuerten    & 4/11   &        &      &         & 0/1     &          &        & 2/3     & 2/5     & 1/4    & 7/12       &         &       &         & 4/7  & 3/3    &       &         &          & 4/8    & 2/4  & 1/2     & 4/7   & 1/3     &          \\
Lendl      & 6/8    & 11/21  & 2/8  & 21/34   & 4/4     &          & 13/27  &         &         &        &            &         &       & 21/36   &      & 4/5    &       & 1/1     &          & 0/1    &      &         &       & 3/8     & 15/22    \\
McEnroe    & 2/4    & 2/10   & 7/14 & 20/34   & 1/3     &          & 7/13   &         &         &        &            &         & 15/36 &         &      &        &       & 6/9     & 1/2      &        &      &         &       & 0/3     & 7/13     \\
Moya       & 1/4    & 2/4    &      &         & 1/3     & 2/4      & 0/1    & 0/7     & 6/14    & 5/12   & 3/6        & 3/7     &       &         &      & 4/8    & 2/8   &         &          & 3/4    & 2/7  & 1/5     & 4/7   & 1/4     &          \\
Muster     & 4/9    & 1/3    &      &         & 5/12    &          & 0/10   &         &         &        & 4/5        & 0/3     & 1/5   &         & 4/8  &        &       &         &          & 0/3    & 3/4  &         & 0/1   & 2/11    & 0/2      \\
Nad      & 2/2    &        &      &         &         & 22/39    &        & 22/32   & 7/9     & 6/10   &            &         &       &         & 6/8  &        &       &         &          &        &      & 7/10    & 2/2   &         &          \\
Nastase    &        & 0/1    & 5/15 & 15/27   & 1/1     &          &        &         &         &        &            &         & 0/1   & 3/9     &      &        &       &         & 4/5      &        &      &         &       &         & 0/1      \\
Newcombe   &        &        & 3/4  & 2/4     &         &          &        &         &         &        &            &         &       & 1/2     &      &        &       & 1/5     &          &        &      &         &       &         &          \\
Rafter     & 5/15   & 1/3    &      &         & 3/3     &          & 0/3    & 3/3     & 1/3     & 1/4    & 2/5        & 4/8     & 1/1   &         & 1/4  & 3/3    &       &         &          &        & 2/3  &         & 1/1   & 4/16    & 1/3      \\
Rios       & 2/3    & 2/5    &      &         & 3/3     &          & 0/1    & 0/2     & 1/4     & 2/5    & 2/8        & 2/4     &       &         & 5/7  & 1/4    &       &         &          & 1/3    &      & 0/2     & 1/4   & 0/2     &          \\
Roddick    & 1/6    &        &      &         &         & 5/9      &        & 3/24    & 5/5     & 7/14   &            & 1/2     &       &         & 4/5  &        & 3/10  &         &          &        & 2/2  &         & 4/7   & 2/3     &          \\
Safin      & 3/6    & 1/1    &      &         & 1/2     & 2/2      &        & 2/12    & 6/12    & 7/14   & 2/4        & 3/7     &       &         & 3/7  & 1/1    & 0/2   &         &          & 0/1    & 3/4  & 3/7     &       & 4/7     &          \\
Sampras    & 20/34  & 12/19  &      & 2/2     & 16/20   &          & 8/14   & 0/1     &         & 4/9    & 11/13      & 2/3     & 5/8   & 3/3     & 3/4  & 9/11   &       &         &          & 12/16  & 2/2  & 1/3     & 3/7   &         & 2/3      \\
Wilander   & 2/7    & 3/10   & 0/1  & 5/5     &         &          & 11/20  &         &         &        & 1/2        &         & 7/22  & 6/13    &      & 2/2    &       & 1/1     &          & 2/3    &      &         &       & 1/3     &      \\
\midrule[1pt]
\multicolumn{26}{l}{\emph{Note:} Due to the space limitation, the names of tennis players have been abbreviated to the first three letters in the horizontal row and their full names can be seen in the vertical column.}\\
\end{tabular*}
\end{sidewaystable}
\restoregeometry

\newgeometry{left=3cm,bottom=2.5cm,top=2cm}
\begin{sidewaystable}
\centering
\caption{Incompete pairwise comparison matrix}
\label{IPCM}

 \tiny
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}llllllllllllllllllllllllll}
\toprule[1pt]
&Aga & Bec & Bor & Con & Cou & Djo & Edb &Fed & Fer & Hew & Kaf & Kue & Len & McE & Moy & Mus & Nad & Nas & New & Raf & Rio & Rod & Saf & Sam & Wil  \\ \hline
Agassi &  & 1.39 &  & 1.07 & 0.90 & &	1.17 & 0.76 & 0.95 & 1.00 & 1.24 & 1.17 & 0.80 & 1.00 & 1.12 & 1.05 & 0.93 &  &  & 1.31 & 0.95 & 1.28 & 1.00 & 0.73 & 1.18   \\
Becker &0.72 &  &  & 1.38 & 1.38 &  & 2.28 &  &  & 1.04 & 1.18 &  & 0.95 & 1.43 & 1.00 & 1.05 &  & 1.03 &  & 1.05 & 1.05 &  & 0.97 & 0.77 & 1.24  \\
Borg &  &  &  & 1.45 &  &  &  &  &  &  &  &  & 1.25 & 1.00 &  &  &  & 1.31 & 0.89 &  &  &  &  &  & 1.03  \\
Connors & 0.93 & 0.73 & 0.69 &  & 0.88 &  & 1.00 &  &  &  &  &  & 0.66 & 0.73 &  &  &  & 0.86 & 1.00 &  &  &  &  & 0.93 & 0.78  \\
Courier & 1.11 & 0.72 &  & 1.13 &  &  & 1.11 &  &  &  & 0.78 & 1.03 & 0.83 & 1.05 & 1.05 & 1.11 &  & 0.97 &  & 0.88 & 0.88 &	 & 1.00 & 0.49 &  \\
Djokovic &  &  &  &  &  &  & & 0.95 & 1.06 & 1.38 &  &  &  &  & 1.00 &  & 0.77 &  &  &  &  & 0.95 & 0.93 &  &   \\
Edberg & 0.85 & 0.44 &  & 1.00 & 0.90 &  &  &  &  &  & 0.95 &  & 1.05 & 0.95 & 1.03 & 1.89 &  &  & & 1.13 & 1.03 & &  & 0.90 & 0.90  \\
Federer & 1.32 &  &  &  &  & 1.05 &  &  & 1.49 & 1.72 & 0.90 & 0.95 &  & & 1.48 & & 0.52 &  &  & 0.88 & 1.07 & 3.31 & 1.64 & 1.03  \\
Ferrero &1.05 &  &  &  &  & 0.95 &  & 0.67 &  & 0.90 & 0.95 & 1.05 &  &  & 1.11 &  & 0.75 &  &  & 1.05 & 1.12 & 0.78 & 1.00 & 0 &   \\
Hewitt & 1.00 & 0.97 &  &  & & 0.72 &  & 0.58 & 1.11 &  & 1.49 & 1.12 &  &  & 1.11 &  & 0.90 &  &  & 1.12 & 1.05 & 1.00 & 1.00 & 1.05 &   \\
Kafelnikov & 0.81 & 0.90 &  &  & 1.28 &  & 1.05 & 1.11  & 1.05 & 0.67 &  & 0.90 &  &  & 1.00 & 0.84 & &  &  & 1.05 & 1.25 & & 1.00 & 0.57 & 1.00  \\
Kuerten & 0.85 &  &  &  & 0.97 &  &  & 1.05 & 0.95 & 0.89 & 1.11 &  &  &  & 1.05 & 1.13 &  &  &  & 1.00 & 1.00 & 1.00 & 1.05 & 0.95 &   \\
Lendl & 1.25 & 1.05 & 0.80 & 1.52 & 1.20 &  & 0.95 &  &  &  &  &  &  & 1.36 & & 1.19 &  & 1.03 &  & 0.97 &  &  &  & 0.90 & 1.54  \\
McEnroe & 1.00 & 0.70 & 1.00 & 1.36 & 0.95 &  & 1.05 &  &  &  &  &  & 0.73 &  &  &  &  & 1.17 & 1.00 &  &  &  &  & 0.88 & 1.05   \\
Moya & 0.89 & 1.00 &  &  & 0.95 & 1.00 & 0.97 & 0.67 & 0.90 & 0.90 & 1.00 & 0.95 &  &  &  & 1.00 & 0.80 &  &  & 1.12 & 0.85 & 0.84 & 1.05 & 0.89 &   \\
Muster & 0.95 & 0.95 &  &  & 0.90 &  & 0.53 &  & &  & 1.19 & 0.88 & 0.84 &  & 1.00 &  &  &  &  & 0.88 & 1.12 &  & 0.97 & 0.65 & 0.93  \\
Nadal & 1.07 &  &  &  &  & 1.29 &  & 1.91 & 1.34 & 1.11 &  &  &  &  & 1.25 &  & &  &  &  & & 1.24 & 1.07 &  &   \\
Nastase &  & 0.97	 & 0.77 & 1.17 & 1.03 &  & &  &  &  &  &  & 0.96 & 0.85 &  &  &  &  & 1.19 &  &  &  &  &  & 0.97  \\
Newcombe &  &  & 1.12 & 1.00 & & & & & &  & & & & 1.00 & & & & 0.84 &  & & &  &  &  &  \\
Rafter & 0.77 & 0.95 &  &  & 1.13 &  & 0.88 & 1.13 & 0.95 & 0.89 & 0.95 & 1.00 & 1.04 &  & 0.89 & 1.13 &  &  &  &  & 1.05 &  & 1.03 & 0.64 & 0.95 \\
Rios &1.05 & 0.95 &  &  & 1.13 &  & 0.97 & 0.93 & 0.89 & 0.95 & 0.80 & 1.00 &  &  & 1.18 & 0.89 &  &  &  & 0.95 &  & 0.93 & 0.89  & 0.93 &   \\
Roddick & 0.78 & &  &  &  & 1.05 &  & 0.30 & 1.28 & 1.00 &  & 1.00 & & & 1.19 &  & 0.80 &  &  &  & 1.07 &  & 1.05 & 1.05 &  \\
Safin & 1.00 & 1.03 &  &  & 1.00 & 1.07 &  & 0.61 & 1.00 & 1.00 & 1.00 & 0.95 &  &  & 0.95 & 1.03 & 0.93 & &  & 0.97 & 1.12 & 0.95 &  & 1.05 &    \\
Sampras & 1.36 & 1.30 & & 1.07 & 2.04 &  & 1.11 & 0.97 &  & 0.95 & 1.77 & 1.05  & 1.11 & 1.13 & 1.12 & 1.53 &  &  &  & 1.57 & 1.07 & 0.95 & 0.95 &  & 1.05  \\
Wilander & 0.85 & 0.80 & 0.97 & 1.28 & & & 1.11 & & & & 1.00 & & 0.65 & 0.95 & & 1.07 && 1.03 & & 1.05  &  &  &  & 0.95 &   \\
\midrule[1pt]
\multicolumn{26}{l}{\emph{Note:} Due to the space limitation, the names of tennis players have been abbreviated to the first three letters in the horizontal row and their full names can be seen in the vertical column.}\\
\end{tabular*}
\end{sidewaystable}
\restoregeometry

\newgeometry{left=3cm,bottom=2.2cm,top=1.8cm}
\begin{sidewaystable}
\centering
\caption{Complete pairwise comparison matrix}
\label{PCM}

 \tiny
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}llllllllllllllllllllllllll}
\toprule[1pt]
    & Aga  & Bec  & Bor  & Con  & Cou  & Djo  & Edb  & Fed  & Fer  & Hew  & Kaf  & Kue  & Len  & McE  & Moy  & Mus  & Nad  & Nas  & New  & Raf  & Rio  & Rod  & Saf  & Sam  & Wil  \\ \hline
Agassi & 1.00 & 1.39 & 0.89 & 1.07 & 0.90 & 1.00 & 1.17 & 0.76 & 0.95 & 1.00 & 1.24 & 1.17 & 0.80 & 1.00 & 1.12 & 1.05 & 0.93 & 1.06 & 1.08 & 1.31 & 0.95 & 1.28 & 1.00 & 0.73 & 1.18 \\
Becker & 0.72 & 1.00 & 0.99 & 1.38 & 1.38 & 0.96 & 2.28 & 0.83 & 1.04 & 1.04 & 1.18 & 1.04 & 0.95 & 1.43 & 1.00 & 1.05 & 0.81 & 1.03 & 1.26 & 1.05 & 1.05 & 1.04 & 0.97 & 0.77 & 1.24 \\
Borg & 1.13 & 1.01 & 1.00 & 1.45 & 1.25 & 1.08 & 1.22 & 0.94 & 1.15 & 1.06 & 1.19 & 1.16 & 1.25 & 1.00 & 1.20 & 1.30 & 0.92 & 1.31 & 0.89 & 1.19 & 1.17 & 1.19 & 1.13 & 0.99 & 1.03 \\
Connors & 0.93 & 0.73 & 0.69 & 1.00 & 0.88 & 0.84 & 1.00 & 0.72 & 0.89 & 0.82 & 0.94 & 0.90 & 0.66 & 0.73 & 0.92 & 1.00 & 0.72 & 0.86 & 1.00 & 0.92 & 0.89 & 0.93 & 0.86 & 0.93 & 0.78 \\
Courier & 1.11 & 0.72 & 0.80 & 1.13 & 1.00 & 0.89 & 1.11 & 0.74 & 0.92 & 0.83 & 0.78 & 1.03 & 0.83 & 1.05 & 1.05 & 1.11 & 0.75 & 0.97 & 1.04 & 0.88 & 0.88 & 0.92 & 1.00 & 0.49 & 0.94 \\
Djokovic & 1.00 & 1.04 & 0.92 & 1.20 & 1.12 & 1.00 & 1.10 & 0.95 & 1.05 & 1.38 & 1.11 & 1.04 & 0.95 & 1.11 & 1.00 & 1.12 & 0.77 & 1.09 & 1.14 & 1.09 & 1.08 & 0.95 & 0.93 & 0.93 & 1.10 \\
Edberg & 0.85 & 0.44 & 0.82 & 1.00 & 0.90 & 0.91 & 1.00 & 0.79 & 0.96 & 0.86 & 0.95 & 0.97 & 1.05 & 0.95 & 1.03 & 1.89 & 0.75 & 0.91 & 0.97 & 1.13 & 1.03 & 0.97 & 0.94 & 0.90 & 0.90 \\
Federer & 1.32 & 1.21 & 1.06 & 1.39 & 1.36 & 1.05 & 1.26 & 1.00 & 1.49 & 1.72 & 0.90 & 0.95 & 1.10 & 1.29 & 1.48 & 1.32 & 0.52 & 1.27 & 1.32 & 0.88 & 1.07 & 3.31 & 1.64 & 1.03 & 1.23 \\
Ferrero & 1.05 & 0.96 & 0.87 & 1.12 & 1.09 & 0.95 & 1.04 & 0.67 & 1.00 & 0.90 & 0.95 & 1.05 & 0.90 & 1.03 & 1.11 & 1.07 & 0.75 & 1.03 & 1.07 & 1.05 & 1.12 & 0.78 & 1.00 & 0.82 & 1.03 \\
Hewitt & 1.00 & 0.97 & 0.95 & 1.22 & 1.20 & 0.72 & 1.17 & 0.58 & 1.11 & 1.00 & 1.49 & 1.12 & 0.97 & 1.14 & 1.11 & 1.17 & 0.90 & 1.10 & 1.17 & 1.12 & 1.05 & 1.00 & 1.00 & 1.05 & 1.13 \\
Kafelnikov & 0.81 & 0.90 & 0.84 & 1.07 & 1.28 & 0.90 & 1.05 & 1.11 & 1.05 & 0.67 & 1.00 & 0.90 & 0.83 & 0.97 & 1.00 & 0.84 & 0.74 & 0.98 & 1.02 & 1.05 & 1.25 & 1.00 & 1.00 & 0.57 & 1.00 \\
Kuerten & 0.85 & 0.96 & 0.86 & 1.11 & 0.97 & 0.96 & 1.03 & 1.05 & 0.95 & 0.89 & 1.11 & 1.00 & 0.89 & 1.03 & 1.05 & 1.13 & 0.78 & 1.02 & 1.06 & 1.00 & 1.00 & 1.00 & 1.05 & 0.95 & 1.03 \\
Lendl & 1.25 & 1.05 & 0.80 & 1.52 & 1.20 & 1.05 & 0.95 & 0.91 & 1.12 & 1.03 & 1.20 & 1.12 & 1.00 & 1.36 & 1.15 & 1.19 & 0.91 & 1.03 & 1.18 & 0.97 & 1.13 & 1.15 & 1.10 & 0.90 & 1.54 \\
McEnroe & 1.00 & 0.70 & 1.00 & 1.36 & 0.95 & 0.90 & 1.05 & 0.78 & 0.97 & 0.88 & 1.03 & 0.97 & 0.73 & 1.00 & 1.00 & 1.10 & 0.78 & 1.17 & 1.00 & 1.01 & 0.96 & 1.00 & 0.93 & 0.88 & 1.05 \\
Moya & 0.89 & 1.00 & 0.84 & 1.08 & 0.95 & 1.00 & 0.97 & 0.67 & 0.90 & 0.90 & 1.00 & 0.95 & 0.87 & 1.00 & 1.00 & 1.00 & 0.80 & 0.98 & 1.03 & 1.12 & 0.85 & 0.84 & 1.05 & 0.89 & 0.99 \\
Muster & 0.95 & 0.95 & 0.77 & 1.00 & 0.90 & 0.89 & 0.53 & 0.76 & 0.93 & 0.86 & 1.19 & 0.88 & 0.84 & 0.91 & 1.00 & 1.00 & 0.74 & 0.91 & 0.94 & 0.88 & 1.12 & 0.93 & 0.97 & 0.65 & 0.93 \\
Nadal & 1.07 & 1.23 & 1.08 & 1.39 & 1.34 & 1.29 & 1.33 & 1.91 & 1.34 & 1.11 & 1.36 & 1.29 & 1.10 & 1.28 & 1.25 & 1.35 & 1.00 & 1.28 & 1.33 & 1.34 & 1.32 & 1.24 & 1.07 & 1.12 & 1.32 \\
Nastase & 0.94 & 0.97 & 0.77 & 1.17 & 1.03 & 0.92 & 1.10 & 0.79 & 0.97 & 0.91 & 1.02 & 0.98 & 0.96 & 0.85 & 1.02 & 1.10 & 0.78 & 1.00 & 1.19 & 1.01 & 0.99 & 0.99 & 0.96 & 0.80 & 0.97 \\
Newcombe & 0.93 & 0.79 & 1.12 & 1.00 & 0.96 & 0.88 & 1.03 & 0.76 & 0.94 & 0.86 & 0.98 & 0.94 & 0.85 & 1.00 & 0.97 & 1.06 & 0.75 & 0.84 & 1.00 & 0.97 & 0.94 & 0.96 & 0.91 & 0.82 & 0.96 \\
Rafter & 0.77 & 0.95 & 0.84 & 1.09 & 1.13 & 0.92 & 0.88 & 1.13 & 0.95 & 0.89 & 0.95 & 1.00 & 1.04 & 0.99 & 0.89 & 1.13 & 0.74 & 0.99 & 1.03 & 1.00 & 1.05 & 1.02 & 1.03 & 0.64 & 0.95 \\
Rios & 1.05 & 0.95 & 0.85 & 1.12 & 1.13 & 0.92 & 0.97 & 0.93 & 0.89 & 0.95 & 0.80 & 1.00 & 0.89 & 1.04 & 1.18 & 0.89 & 0.76 & 1.01 & 1.07 & 0.95 & 1.00 & 0.93 & 0.89 & 0.93 & 1.00 \\
Roddick & 0.78 & 0.96 & 0.84 & 1.08 & 1.08 & 1.05 & 1.03 & 0.30 & 1.28 & 1.00 & 1.00 & 1.00 & 0.87 & 1.00 & 1.19 & 1.07 & 0.80 & 1.01 & 1.04 & 0.98 & 1.07 & 1.00 & 1.05 & 1.05 & 1.00 \\
Safin & 1.00 & 1.03 & 0.88 & 1.16 & 1.00 & 1.07 & 1.06 & 0.61 & 1.00 & 1.00 & 1.00 & 0.95 & 0.91 & 1.07 & 0.95 & 1.03 & 0.93 & 1.04 & 1.10 & 0.97 & 1.12 & 0.95 & 1.00 & 1.05 & 1.05 \\
Sampras & 1.36 & 1.30 & 1.01 & 1.07 & 2.04 & 1.07 & 1.11 & 0.97 & 1.22 & 0.95 & 1.77 & 1.05 & 1.11 & 1.13 & 1.12 & 1.53 & 0.89 & 1.25 & 1.22 & 1.57 & 1.07 & 0.95 & 0.95 & 1.00 & 1.05 \\
Wilander & 0.85 & 0.80 & 0.97 & 1.28 & 1.06 & 0.91 & 1.11 & 0.81 & 0.97 & 0.88 & 1.00 & 0.97 & 0.65 & 0.95 & 1.01 & 1.07 & 0.76 & 1.03 & 1.05 & 1.05 & 1.00 & 1.00 & 0.96 & 0.95 & 1.00 \\
\midrule[1pt]
\multicolumn{26}{l}{\emph{Note:} Due to the space limitation, the names of tennis players have been abbreviated to the first three letters in the horizontal row and their full names can be seen in the vertical column.}\\
\end{tabular*}
\end{sidewaystable}
\restoregeometry

Firstly,
the statistical competition results matrix (i.e. with additive preference relations) is converted into the pairwise comparison matrix (i.e. with multiplicative preference relations) following the method described in \cite{bozoki2016application}.
Fig.\ref{transformation} illustrates this transformation process. The converted pairwise comparison matrix is shown in Tab.\ref{IPCM}.

Secondly, DEMATEL-based completion method is adopted to complete the pairwise comparison matrix. Following the procedure of the proposed method, the complete pairwise comparison matrix is obtained, which is shown in Tab.\ref{PCM}.

Thirdly, based on the complete pairwise comparison matrix, the priorities of tennis players can be calculated via (\ref{weights of AHP}).
Tab.\ref{Ranks} shows the final ranking of the 25 tennis players.

\begin{table}[htbp]
\centering
\caption{Ranking of 25 tennis players}
\label{Ranks}
\scriptsize
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}clcclc}
\toprule[1pt]
\textbf{Rank} & Player   & Priority \quad & \textbf{Rank}  & Player     & Priority \\ \hline
\textbf{1}    & Nadal    & 0.0503 & \textbf{14}   & Rios       & 0.0380 \\
\textbf{2}    & Federer  & 0.0503 & \textbf{15}   & McEnroe    & 0.0380 \\
\textbf{3}    & Sampras  & 0.0467 & \textbf{16}   & Nastase    & 0.0380 \\
\textbf{4}    & Borg     & 0.0444 & \textbf{17}   & Rafter     & 0.0378 \\
\textbf{5}    & Lendl    & 0.0436 & \textbf{18}   & Wilander   & 0.0378 \\
\textbf{6}    & Becker   & 0.0430 & \textbf{19}   & Kafelnikov & 0.0375 \\
\textbf{7}    & Hewitt   & 0.0414 & \textbf{20}   & Edberg     & 0.0374 \\
\textbf{8}    & Djokovic & 0.0412 & \textbf{21}   & Moya       & 0.0371 \\
\textbf{9}    & Agassi   & 0.0409 & \textbf{22}   & Newcombe   & 0.0365 \\
\textbf{10}   & Safin    & 0.0392 & \textbf{23}   & Courier    & 0.0360 \\
\textbf{11}	  & Kuerten  & 0.0390 & \textbf{24}   & Muster     & 0.0353 \\
\textbf{12}   & Roddick  & 0.0384 & \textbf{25}   & Connors    & 0.0339 \\
\textbf{13}   & Ferrero  & 0.0383 &      &            &    	\\
\bottomrule[1pt]
\end{tabular*}
\end{table}

\section{Discussion}

The result in last section indicates that the proposed method has ability to complete the pairwise comparison matrices in AHP.
In this section, we would further analyze the superiority and efficiency of the proposed method.

The proposed method has the following advantages:
\begin{itemize}
\item
The proposed method estimates the missing values in pairwise comparison matrix from a new perspective (i.e. by calculating the total relations among alternatives).
\item
The proposed method has low computational cost.
\end{itemize}

After reviewing the prior studies, an assessment for the computational cost of the existing completion methods and the proposed method is done. The methods for comparison satisfy:

\begin{itemize}
\item
The methods belong to completion methods, which have ability to estimate the missing values in pairwise comparison matrix.
\item
The pairwise comparison matrices, which can be completed by these completion methods, present reciprocal preference relations among alternatives. In other words, the matrices request that the multiplication of each pair of symmetric values along the diagonal line is equal to 1.
\end{itemize}

As for these completion methods, some of them estimate the missing values through iteration, or based on the complex theories/methods, which causes the methods expensive computational cost.
For example,
Bozo\'iki \emph{et al.}\cite{bozoki2010optimal} developed a completion method which estimates the missing values in pairwise comparison matrix through iteration and optimizes the matrix consistency  based on graph theory.
Gomez \emph{et al.}\cite{gomez2010estimation} proposed a completion method based on the neural network. Neural network as one of the branches of artificial intelligence leads to the complexity of the completion method.
As for the completion methods without iteration or complex theories/algorithms such as the methods listed in Tab.\ref{comparative methods}, time complexity is used to accurately reflect their computational cost. Assume there is an $m \times m$ incomplete pairwise comparison matrix, which has n missing values, the time complexity of these methods is shown in Tab.\ref{timeComplexity}.

\begin{table}[htbp]
\centering
\caption{Methods for comparison}
\label{comparative methods}
\scriptsize
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}lll}
\toprule[1pt]
&  	\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad Description &  \quad \quad Source 	\\ \hline
Method 1    &	\tabincell{l}{
A completion method extends the geometric mean induced bias matrix to est-\\~~imate the missing judgments and improves the consistency ratios by the least\\~~absolute error method and the least square method.
}
&  Ergu \emph{et al.}\cite{ergu2016estimating} \\
Method 2 	&  	\tabincell{l}{A consistency-based completion procedure, which takes a measure to maintain \\ ~~the consistency level of matrices before and after completion.} 	& Alonso \emph{et al.}\cite{alonso2008consistency}  \\
Method 3 	&  \tabincell{l}{A completion method to obtain the most consistent/similar matrix with the in-\\~~itial incomplete matrix based on Frobenius norm.} &  Ben\'itez \emph{et al.}\cite{benitez2015consistent} \\
\bottomrule[1pt]
\end{tabular*}
\end{table}

\begin{table}[htbp]
\centering
\caption{Time complexity of methods}
\label{timeComplexity}
\footnotesize
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}lcccc}

\toprule[1pt]
					& Method 1 			& Method 2  			& Method 3 		& The proposed method \\
\hline
Time complexity		& {$m^3$}			& {$m^3$}		& {$m^2+mn$}			& {$m^2$}				\\
\bottomrule[1pt]
\end{tabular*}
\end{table}

It can be seen from the Tab.\ref{timeComplexity} that the time complexity of the proposed method is lower than the others.
In addition, the most existing completion methods have the following two operations:
\begin{itemize}
\item Divide the matrix into upper and lower triangular matrix.
The pairwise comparison matrix in AHP has a property that the multiplication of each pair of symmetric values along the diagonal line is equal to 1. These methods firstly focus on the completion of upper/lower triangle of incomplete matrix and use the property to accomplish the other triangular matrix.
\item Traverse the pairwise comparison matrix to obtain the positions of unavailable values, then estimate these missing values in sequence.
\end{itemize}

However, the proposed method completes the matrix from a holistic viewpoint without dividing the matrix into the upper and lower triangular matrix.
Furthermore, based on DEMATEL, all the missing values in matrix can be simultaneously estimated.
Hence, following the procedure of the proposed method, the completion of incomplete pairwise comparison matrix becomes straightforward and efficient.

\section{Conclusion}
\label{conclusion}
Pairwise comparison matrix plays a pivotal role in AHP. However, in many cases, only partial information in pairwise comparison matrix is available, which obstructs the subsequent operations of the classical AHP.
In this paper, we propose a new completion method for incomplete pairwise comparison matrix in AHP.
Based on DEMATEL, the proposed method has ability to complete the matrix by obtaining the total (direct and indirect) relations from the direct preference relations between each pair of alternatives.
Furthermore, an application of the proposed method for ranking tennis players is presented.
From the ranking result and the comparison with several representative completion methods,
the proposed method is proved to be effective and efficient.
The proposed method provides a new perspective to complete the matrix with explicit physical meaning. Besides, the proposed method has low computation cost.
This promising method has a wide application in multi-criteria decision-making. In our further study, we would extend the proposed method for the incomplete fuzzy pairwise comparison matrices.

\section*{Acknowledgment}

The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61573290,61503237), China State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant No.BUAA-VR-14KF-02).

\section*{References}

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redpajama_set_name: RedPajamaArXiv


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Even though we finished classes and exams last Wednesday before the four day Easter weekend (Friday-Monday) our school officially closes today. This means kids will go home for term break. All kids except 40 who will stay for the regional sport competition held at our school.
Tomorrow, 80 kids and about 20 teachers from two other schools for the deaf will arrive. Then we have sport events Thursday-Saturday and they'll leave Sunday morning. It should be a busy week but hopefully fun!
During the last presidential election in Kenya there was violence which resulted in the evacuation of Peace Corps Volunteers. This time around Peace Corps wanted to be prepared in case we needed to be evacuated for this year's presidential election. All 110 PCVs met up in a nice remote area for consolidation. We spent about a week and half there. It's not often that PCVs get to meet so many other PCVs in their country so it was a great experience! For the most part, things were calm across the country so we got to return to our sites via Peace Corps vehicles (was SO nice not having to drag all of our things across Kenya via public transportation).
The week prior to and after consolidation I think was stressful for most of us. We didn't know if we would ever return to our villages again, we didn't know if we should say good-bye to our students/community members, we didn't know if we would get to see loved ones in the States who we hadn't seen in over a year, we didn't know if we would be living in the States or Kenya in just a few weeks, we didn't know if we would have a job in just a couple of weeks…..then there was the packing! We had to make an inventory of everything in our house, pack things up to the point where if we were evacuated then certain things we left behind would be shipped to us yet not pack everything up in case we did come back, we had to make a few piles (things to be shipped, things to give to people, things to take with us to consolidation, PC properties, etc). Then once we returned to sites we had to unpack everything, get settled back into things, come to terms that we were going to be able to continue our job and projects in Kenya, and readjust to Kenya culture after getting to be ourselves without judgement around other Americans for a week and half.
Things are great now! I'm back into the flow of things and am glad to be back at my site with my kiddos! It's crazy though, the first term is coming to an end already!! Final exams should be this coming week then we will start the annual sports events that occur during the first term.
Hopefully I'll do a bit better about blogging. It's kind of hard now because I've forgotten what isn't "normal" to Americans. What was at one point new and unusual to me is part of my daily life and I don't think about it as much anymore.
Cooking can easily take all day in Kenya if you cook Kenya food. Every once in a while I make chapatis but that's usually the only Kenyan food I make..
My second host sister teaching me how to make chapatis on a jiko stove (my first host family had a family emergency so I ended up getting moved to a second host family during training).
If you want green grams or beans with your meal then you need to soak them in water the night before so that they soften up enough to cook.
Same with maize (corn), they need to sit out in the sun to soften up.
Like most PCVs, I stick with simple and food that's quick to cook (e.g. spaghetti…I've never eaten so much spaghetti in my life before). I also usually add Sossi which is soya (good source of protein) since it's not exactly sanitary (by my standards) or easy to purchase meat in my village. Meat is usually very tough and sits out in the sun for weeks until it's all sold. My school actually spent about 2-3 weeks eating meat from a cow that had to be killed due to an illness (no refrigerator or freezer….just tree shade).
Kenyans may cook outside or inside with gas, kerosene, charcoal (jiko stove) or firewood.
Most people in my village do not use gas tanks. There aren't any gas pipes that lead to houses, instead people travel to town to purchase gas tanks which can be a pain to transport back. My first host family lived in town so they used a gas tank and a jiko stove. My second host family lived out in the village and used wood to cook and a jiko stove. They would chop down tree branches and carry them to the house on their back.
My second host family's kitchen. The red cup is on top of a pot which is where the stove is.
My first host family's kitchen. The gas tank is black, most tanks I've seen are about half this size.
I use kerosene which is around 80-90/= per litre, lasts for a while (around 400/= worth lasts a few months), and easy to transport.
I use a kerosene stove. I light up all 8 wicks (it's a pain to switch out the wicks when they get short) and hope a flame ball doesn't shoot up. Sometimes I'll be cooking something before I realize I've just about ran out of kerosene ….this happens every once in a while and ends up doubling my cooking time. I only have one stove so if I want to make spaghetti with white sauce and garlic bread then I time it so that I can still have warm food to eat after cooking one item at a time…it does take longer cooking one item at a time instead of all at once. The downside to kerosene is when you get the smoke in your eyes and it's a pain to clean the bottom of pots….ugh!
Yellow containers are reused cooking oil containers (Kenyans use SO much cooking oil) which are resold to be used as kerosene containers or to transport water. You see them all over the place in Kenya. When I need to get kerosene I wait 1-2 hours by the side of my dirt road for a vehicle, travel about an hour to town, hope I can get kerosene at the first gas station I go to (sometimes stations run out of kerosene), if the gas station is busy then I have to push my way through and stick the yellow container in front of the person who is doing the filling (they have an employer who does it and there is no concept of lines or going in order in Kenya….it's all about pushing your way through) then it takes 0.5-4 hours to get a vehicle to leave town, and another hour back to the village. I have spilled kerosene all over my pants before while traveling on the bumpy dirt road….am just glad no one near me lit a match!
One has to make sure they don't run out of gas or kerosene completely….which is so easy to do. If I'm low on kerosene and I know I won't be able to make it to town for a few days I'll either eat a lot of PBJ sandwiches or something that's very quick to cook (e.g. scrambled eggs).
It's Monday which means time for another "How Do I…." post?!
I typically throw dishes and pots into a basin that's 1/3 full of water since I don't have running water or a sink. This basin is in my "washroom" which is just an empty room with a small drainage that goes outside and is where I do all of my cleaning at (bathing, washing dishes, washing clothes, etc). Yes, the basin is on the floor….lots of things get done on the floor around here (bathing, cooking, washing dishes, washing clothes).
I then add dish soap (by the time I get a vehicle to go to town and return back to my village…it can take 4-6 hours to get this bottle of soap) to the basin.
Oftentimes the weather affects what I cook. If we're in a dry spell then I'm more aware of what I cook. For example I may eat a PBJ sandwich and a boiled egg for one meal which requires minimal water to cook and clean up. I may choose to make spaghetti instead of pancakes…again, less water needed to clean up afterwards.
I use a small pitcher to clean pots since it's a lot easier to use less water that way instead of filling up the basin full of water.
If I've had an easy lunch (e.g. PBJ sandwich) then I just reuse the water for clean-up after dinner if it's clean enough. Since I have cockroaches, bats, spiders, etc. crawling around I wash my plate and fork right before I put my food on it even though I just washed them the day before.
Then, depending on what was cooked….I either sweep or mop up the floor! I was able to grab this broom a while back from town (vs the typical Kenya broom which I do use from time to time). I later found a mop attachment which didn't fit and the broom attachment wouldn't come off so I ended up whittling down one end of the broom to attach the mop….hey…it works.
How Do I…..Use the Restroom?
A choo is an outhouse. I do not have my own choo, so I share one with the headmaster, boys housemama, deputy headmaster, and the headmasters' housegirls (it is common for middle class Kenyans to have housegirls who will clean, take care of young ones, etc.) The choo is at the other end of the school campus from my house.
The door on the left is the choo and the blue door on the right is the washroom for the headmaster/deputy headmaster/boys housemama/etc) (lucky I have a room in my house that I can use as a washroom).
Where you squat down and do your business in a hole. Our choo is usually clean most of the time (except when we have visitors) since the kids are responsible for cleaning the choos on a daily basis. No one can ever aim into the hole…..just sayin'. On bad days, there will be flies that fly up and hit your behind….so glad that doesn't happen often though. There are geckos that live in there, they scurry away whenever I open the door. So far I haven't seen any snakes but I do have to keep that in mind and do a quick snake check before I enter.
Our school is wonderful about cleanliness…the kids even bath twice a day. We also have washing stations like this one outside of the choo. During the dry season we keep the water that ends up in the basin and it's either put back in the blue container to be used again (if it's soapy enough) or used to mop classroom floors.
One tip if you ever visit Kenya…always, always, always carry toilet paper with you. They're not provided in choos and most public restrooms do not provide them either….


text:
Valuable Books and Manuscripts
SMITH, Adam (1723-1790). Autograph letter signed ('Adam Smith') to [William Eden, later 1st Baron Auckland], Edinburgh, 9 December 1783.
2½ pages, 253 x 195mm (the last six lines cut down from originally conjoined f.2 and now on a loose fragment, approx. 90 x 187mm; a smaller fragment with the phrase 'Commissioner of the Customs' pasted onto the foot of f.1v; f.1 reinforced at left and lower margins, with traces of mount). Provenance: sale at Bonham's, London, 18 June 2014, lot 168.
'... our future Commercial connexions with our thirteen revolted Colonies ...': on trade with America, preventing smuggling, and not being 'the only Adam Smith in the world'. Writing as commissioner of customs in Edinburgh, a post he had occupied since 1778, Smith refers to some accounts which he had intended to send to Eden, explaining that although 'the Officers whose business it is to prepare them had given me reason to expect that they would be ready before this time', they will now be delayed for another few days. He goes on 'The report of the board of Customs here, concerning the proper method of preventing smuggling, is likely to be so perfectly agreable [sic] to my own ideas, that I shall not anticipate it by giving you any account of them'. As soon as the customs board is adjourned, Smith will write again to 'endeavour to answer as fully and distinctly as I can, all the questions you have done me the very great honour to ask me concerning our future Commercial connexions with our thirteen revolted Colonies'. A postscript notes that Smith is likely to be appointed by the customs board as commissioner to Eden's parliamentary committee: he continues on a humorous note 'When you do me the honour to write to me, Be so good as to direct to me Commissioner of the Customs. I once had the vanity to flatter myself that I was the only Adam Smith in the world; but to my unspeakable mortification, there are two or three others of the same name in this town, and my letters have sometimes gone wrong'.
The independence of the United States of America had been recognised by Great Britain only three months previously, at the Treaty of Paris on 3 September. The recipient, William Eden (1745-1814, created Baron Auckland in 1793), had written to Smith in connection with the American Intercourse Bill, asking his advice on the wisdom of according a free trade treaty to the new republic, and expressing fears as to the potential impact on English, Canadian and Irish commerce. Smith replied (on 15 December) strongly opposing any restrictions, though more on the grounds of equality of treatment than strictly of free trade. In spite of his hesitation on this point, Eden was strongly influenced by Smith's ideas both in 1779, when as secretary of the Board of Trade he removed trade barriers between Ireland and both British and colonial markets, and again in 1786, when he negotiated a commercial treaty with France which favoured free trade. The reference to smuggling in the present letter is a reminder of the irony of Smith's responsibility as commissioner of customs for suppressing such activities, in spite of his defence of the principles of smuggling in Wealth of Nations. Not published in The Correspondence of Adam Smith (ed. Mossner and Ross, 1987).
Sophie Hopkins
SHopkins@christies.com +44 (0) 207 752 3144
More from Valuable Books and Manuscripts

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communication technology degree salary
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Kannapolis is home to more than 42,000 people. Get on a bus charter in Kannapolis, North Carolina and see the attractions in the city.
Gem Theater is where you could watch the latest movies and the best events such as the NC Music Hall of Fame Induction Ceremonies, the Summer Movie Camp, and the Modern Film Festival. You should also go to the Curb Museum for Music and Motorsports, where you could learn more about NASCAR racing and music. You can see cars that were driven by Richard Petty, Ron Bouchard, and Dale Earnhardt, among others. Then check out the North Carolina Music Hall of Fame, which is also located on the site.
Play golf at Club At Irish Creek and get 18 holes of fun over more than 6,800 yards at the longest tees. The course has a par of 72, a rating of 70.4 and a slope rating of 127. Designed by both Monroe Maulden and Davis Love, the course first opened in 2007. Meanwhile, you can also check out the CMC - NorthEast Stadium, which seats around 4,700 people and is the home of the Kannapolis Intimidators.
Visit the Meek House, which was built in the 1830s and has a combination of Greek revival and federal architecture. It stands on land that measures close to 14 acres. Or you could check out the Harvey Jeremiah Peeler House, built in the 1920s and has a colonial revival and American craftsman architecture. It also became the site of Lady's Funeral Home until the late 1960s. Both of these houses are listed on the National Register of Historic Places.
There are several parks that you can go to in Kannapolis. Veterans Park has beautiful green spaces that feature fountains and brick paths. This is a great venue for special events and concerts. See the eternal flame, Memorial Plaza monument and Ring of Honor here. The Dale Earnhardt Plaza, on the other hand, has one acre of space and pays tribute to the racing legend. Find the Earnhardt bronze statue, which stands nine feet into the air surrounded by gardens and paths.
Meanwhile, the Village Park is host to the Summer Entertainment Series at the amphitheater. The series draws more than 45,000 people every year. You can also find the Splash Pad here and enjoy the geysers, water jets, and water cannons. Further, ride the Rotary Express Train or check out what's happening at the Village Park Building. Then you have Bakers Creek Park, which has 40 acres of space and features picnic facilities, a playground, mini golf course, horseshoe pits, tennis, volleyball and basketball courts, and a softball field.
Furthermore, you have Walter M. Safrit Park, which has 17 acres of space that features athletic fields and sports courts, a playground, and a walking trail. Whereas Vietnam Veterans Park has 90 acres of space that includes playgrounds, picnic facilities, volleyball and basketball courts, bench swings, a disc golf course, horseshoe pits, a boardwalk, a walking trail, a wetlands trail, and bocce courts.
NASCAR fans simply must go on the Dale Trail, which takes you to the places that have something to do with the life and career of Dale Earnhardt.
To get a free online quote for a bus charter in Kannapolis, North Carolina just call 1-866-569-2012.
Be sure to visit BusRental.com and request a free online quote for a bus charter in Kannapolis, North Carolina.


What is the closest rocket you can dodge?
Depends on which direction you're facing. My understanding is that the player hitbox is always aligned to the map, so the hitbox presents a wider target when shooting at an angle other than normal to a face of the box.


text:
Fan News
Crisis Management and Other Unexpected Lessons from the Batcave
Crisis Management and Other Unexpected Lessons from the...
By Meg Downey Tuesday, September 12th, 2017
What's Batman's most appealing quality? His fighting skill? His endless supply of gadgets? In this ALL STAR BATMAN VOL. 2 breakdown, Meg Downey suggests something far less expected…and far more human.
Hi, my name is Meg and I'm a Batman fan.
Now, okay. You probably had one of two reactions reading that. You either thought "great, me too!" or "ugh, him?"
If you fell into that second category, don't worry. I've been doing this for a while—writing about super heroes, that is—so I know how it goes. Batman's a pretty popular guy, after all, and sometimes that popularity can actually work against him. It's easy to see the yellow-on-black symbol or the scalloped cape or the horned cowl and think you've got this guy all figured out. I mean, he's existed for nearly a hundred years. He's the guy who always knows what to do, who always can afford whatever he needs, who always has a plan. He's simultaneously the least powerful and strongest member of the Justice League. He's Batman.
But hear me out on this one, okay?
I want to talk about a different kind of Batman—the kind who maybe isn't as all-powerful and impossibly well prepared as his reputation makes him out to be. The kind who might actually be trying to set his shoulders and keep his knees from buckling under the weight of a million different ever looming anxieties.
The kind of Batman that gets explored in ALL STAR BATMAN VOL. 2: THE ENDS OF THE EARTH.
People know Batman as the sort of guy who is quite literally prepared for anything, and for the most part, that's pretty true. By and large, he's ready to face whatever challenges are thrown at him from just about every angle. That's part of what makes him seem so invulnerable. But the truth is, the thing that makes Bruce who he is? The thing that keeps him victorious time and time again? The plans are only a part of it, and a small part at that.
The real thing that keeps Batman going, pushing through the weight on his shoulders? It's not actually that special, and it's certainly not something he bought or made with his money or his training. It's the singular, bold, brilliant choice to look for and accept help when he needs it most.
In Ends of the Earth, there are four stories at play, and all four of them have to do with the world coming to an end. (Is that title a bit more ominous than you were expecting?) Whether by an apocalyptic event, or a biological plague, or a corruption of technology, the fate of humanity hangs in the balance. Time for someone like Batman to shine, right?
Well, yes and no. The thing about these stories is that, while Bruce might be ultimately victorious (spoiler alert: the earth doesn't actually end), his course is far from easy, and we get a look into just how challenging it all actually is thanks to the narration of each story. Writer Scott Snyder employed three different styles and perspectives in the first three issues: third person "storybook" style, first person, and second person, all of which culminated in the final chapter—a classic comic book showdown. In doing so, we get a front row seat to the inside of Batman's head as he works...and it's not always as neat and tidy as you might expect.
In fact, it's actually a little...well, chaotic.
Make no mistake, Batman's still Batman, he's still as on top of his game as he could possibly be, but this is a side of him we don't usually get to see, at least not in such glaring detail. It's a little bit like forcing him to stand in the center of a fold-out mirror: four different angles showing four different sides of the same thing. And through those different sides, we get to see that Batman's gut reaction… You know, the anxiety stirred up in the back of his mind? It looks awfully familiar.
The world is a scary place and it keeps finding new ways to get scarier. It's difficult to even look at the news in passing without running into some new potential apocalypse. And really, overcoming the end of the world is something super heroes are, by and large, pretty good at. They've been circumventing catastrophe for almost a hundred years. And it's comforting, it really is, to watch them triumph again and again. I like watching the good guys save the day.
But there's something even more comforting about getting to know that, even for a split second, the heroes who are doing the heavy lifting are just as vulnerable as you are. They're just as nervous, just as anxious, just as in need of a helping hand…and they push through and save the day anyway.
Batman might be the guy who always has the answers at the end of the day, but that's not why I like him. I like him because those answers come from an endless push and pull, outside, inside, upside down. It's a cycle of worry, insecurity and abject anxiety that I recognize all too well. But, hey, if Batman can push through it and literally save the entire planet, I can probably push through it and make it through my completely normal life, right?
Meg Downey writes about Rebirth for DCComics.com and covers Legends of Tomorrow for the #DCTV Couch Club. Look for her on Twitter at @rustypolished.
ALL STAR BATMAN VOL. 2: ENDS OF THE EARTH by Scott Snyder, Jock, Francesco Francavilla, Tula Lotay and Giuseppe Camuncoli is now available in print and as a digital download.
Rebirth Reflections
The Heart of the Matter: Batman's Rebirth Road So Far
Deathstroke vs. Batman: Sins of the Father(s)
Titans: Omen Earns her Name
How to be (In)Human: Lessons from Young Monsters in Love
Nightwing: Blockbuster is Back Again
The Demon: Hell is Earth Offers Etrigan a Fiery Return
Batwoman: Redefining the Familiar
Titans: Do It for the Annual
Wonder Woman: Stretching Belief
Superman Can Always Save You
The Flash: Between Heroes and Rogues
New Super-Man: Why I Love Peng Deilan, the Wonder-Woman of China
Teen Titans: Never too Young for a Rebirth
Trinity: The Strength of Emotions
Supergirl: Chasing the Idea of Home
The Flash: Silly Rogues and Secrets
The Flash Vol. 2 Races Through the Dark
Wonder Woman: Facing Reality Is Overrated
Batgirl and the Birds of Prey: Fandom Turned Frightening
Titans: Whatever Happened to Wally West?
Wonder Woman: The Story Keeps Changing
Godspeed to Barry and the Kid Flashes
Frank Miller's Dark Knight is Back
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Ponyo - Studio Ghibli Fest 2022 - Japanese with English Subtitles
Limited May 16, 2022
When Sosuke, a young boy who lives on a clifftop overlooking the sea, rescues a stranded goldfish named Ponyo, he discovers more than he bargained for. Ponyo is a curious, energetic young creature who yearns to be human, but even as she causes chaos around the house, her father, a powerful sorcerer, schemes to return Ponyo to the sea. Miyazaki's breathtaking, imaginative world is brought to life with an all-star cast. Featuring the voices of Cate Blanchett, Matt Damon, Lily Tomlin, Liam Neeson, and more.


Welcome to Slater Ridge, a mariner's paradise nestled in Slater Farms.. Stroll alongside the seven-acre lake by day, and marvel at our stunning lit fountains by night. This friendly new home community in Noblesville has easy access to the pool, the clubhouse, and the park. Relax under the covered picnic area, burn off some steam on our sports court, or experience a beautiful bike ride on the trail running throughout the community. Rural terrain with impressive estates – that's what you'll find at Slater Ridge. Your new home awaits!
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Diamondbacks 6, Dodgers 4: In play, run(s)
By Jim McLennan@AZSnakepit Jul 31, 2011, 2:15am EDT
Share All sharing options for: Diamondbacks 6, Dodgers 4: In play, run(s)
Record: 58-49. Pace: 88-74. Change on last season: +18.
Watching a game on one of the online apps is a very binary experience. Stuff happens, or it doesn't. There's not the same sense of uncertainty you get actually watching a game. Was that a ball or a strike? Is it a fair ball? Safe or out? Things simply pop up on your screen as undisputed facts, provoking immediate happiness or sadness. The only area of uncertainty is the titular line: In play, run(s). Whether a batter has a sacrifice fly or a grand-slam, they all get the same. But I can't say I've appreciated those three little words much more, than when they popped up on my screen during the top of the sixth inning.
Tthere is absolutely no reason to throw Either or Kemp anything that's anywhere near the strike zone. Ever.
-- Zavada's Moustache
In the first two games of this series in Los Angeles, Andre Ethier and Matt Kemp have combined to go 8-for-17 with a couple of homers and eight runs driven in. S'funny how what can seem so obvious to us, sitting on the sidelines, is apparently an arcane and inexplicable mystery to Arizona's pitching staff. Instead, our starting pitchers have served up so many meatballs to the pair, Kemp and Ethier must think they're in Ikea. Tonight's purveyor of delicacies, Micah Owings, seemed intent on combining with Josh Collmenter yesterday, with the aim of leaving me glad for the arrival of Jason Marquis.
Owings had a very easy first, retiring Los Angeles in order in only nine pitches. The second was a little stickier, as Matt "inevitable" Kemp singled and stole second, but Owings escaped that frame intact, leaving Dodgers on the corners. He couldn't escape the third, however, as a lead-off triple came across to score on another sacrifice fly. Worse was to follow as the despicable duo of Ethier and Kemp singled and homered respectively with two outs, and the Arizona advantage was mere dust in the wind. Another run crossed home for Los Angeles in the fourth, to make the score 4-1 for the home team.
Chad Billingsley certainly looked hittable early on. Two knocks in the first, two more and a walk in the second, but the Diamondbacks couldn't capitalize beyond that, the only hit in six attempts with runners in scoring position over the opening couple of frames, failing to score a run. Particularly galling was the bases-loaded no outs situation Arizona worked in the second, where all we managed was a Willie Bloomquist sacrifice fly, between an Owings K and a Kelly Johnson fly out. Still one run is better than none, and the Diamondbacks did have the lead. However, subsequent events would lead to ruing those wasted opportunities.
Unfortunately, Billingsley settled down, retiring 12 of the next 14 batters that he faced, from when we'd loaded the bags with no outs in the second, through the end of the fifth inning We did get a man into scoring position in the fourth, but Bloomquist could only ground out back to the mound. The sixth inning proved a return to earlier form. Chris Young singled and took second on the next pitch, followed by walks to Brandon Allen and Ryan Roberts, and for the second time in the game, Arizona had the bases loaded with nobody out. Could they take better advantage of the change this time round?
Well, Gerardo Parra immediately made sure they wouldn't do any worse, drawing a walk on a full count to move the tying run in to scoring position. Kirk Gibson then made the decision to replace Owings with Xavier Nady, who did better than many people probably expected, getting another run in with a sacrifice fly, and moving that all-important tying run up to third, still with only one out. The sane from Bloomquist, just liked he'd done in the second inning, would have tied the content, but it was not to be, as he took strike three. Kelly Johnson walked, re-loading the bases for Upton.
That was the end of Billingsley, and the Dodgers went to their bullpen, sending up Matt Guerrier for the sole purpose of facing J-Up. This was an ideal opportunity for our young star to do something productive with the aggression seen in the dugout late yesterday, by hammering one in to the gap, or better yet, into the bleachers. Or would he let his impetuosity get the better of him again, providing more fodder for the nattering nabobs of AZCentral? If the first swing suggested the latter, the title of the recap should have given you a clue to the outcome. A beautiful, bases-clearing double to right-center, and Arizona were back in front, 6-4.
On the other hand, this did mean it was over to the Diamondbacks bullpen, with a lead to protect for 12 outs. Ryan Cook, freshly off the bus from Reno to replace the ailing Alberto Castillo (elbow tendinitis, thank you for asking) was first up. He did get two outs, but a single and a walk led to his departure, in favor of UberLOOGY Joe Paterson, now restored to his position as our sole left-hander. He got pinch-hitter James Loney to ground out, and the Arizona Diamondbacks had steered themselves safely through the sixth-inning shoals. The seventh inning was similar. A lead-off single, a mid-inning change - this time, Bryan Shaw coming in - and no damage done.
The eighth saw Arizona waste some more chances, getting two men in scoring position with no outs, thanks to a pair of singles, followed by a wild pitch. But Kelly Johnson struck out and, after an intentional walk to J-Up [the Dodgers pitching staff having finally decided, albeit too late, to adopt the approach espoused by Zavada's Moustache], Miguel Montero hit into a double-play and the D-backs were unable to pad their lead. However, David Hernandez threw a 1-2-3 ninth to put Arizona on the brink.
After two easy fly-outs, closer J.J. Putz couldn't quite put away Tony Gwynn, Jr - though was not helped by at least one dubious call from Cowboy Joe West at home. An 11-pitch walk brought the tying run to the plate, with the ominous shapes of Ethier and Kemp lurking on-deck and in the hole. Fortunately, J.J. Putz decided to spare our lower colons further punishment, and another fly-ball ended the game, bringing Arizona back to three games behind San Francisco.
I think that's an outcome few would have expected when we went 4-1 down, but Justin Upton continues to prove his MVP credentials. He went 2-for-4 with a walk and three RBI, extending his hitting streak to 11 games. Over that time, he is a ridiculous .500, having gone 21-for-42, with 14 of those hits being for extra bases, including six home-runs, and 19 runs driven in. His OPS coming in was a decent .846, but it's now risen all the way to .933, in less than two weeks. If he keeps this up, Sprankton might run out of up-arrows during next week's Ups and Downs.
[Click to enlarge, at fangraphs.com]
Leonardo da Vinci: Justin Upton, +42.2%
A melted watch-load of Salvador Dalis: Parra, +17.1; Shaw, +14.2%; Allen, +11.6%; Roberts, +10.8%
ToPortraits-Studio: Micah Owings, -26.2%
A load of Jackson Pollocks: Bloomquist, -14.2%, Montero, -13.0%
Only 521 comments in the thread, but it was a late game on a Saturday, and I'm sure everyone had better things to do. Except me... :-( Clefo snagged top spot on 78, with the laideez representing well, 'charmer, 'claws and 'starr12 all getting over 50 [comments in the girls' thread not included]. Also present: 4 Corners Fan, Jim McLennan, porty99, BulldogsNotZags, blank_38, sonic barracuda, rfffr, Sprankton, emilylovesthedbacks, njjohn, dbacksfann, Dallas D'Back Fan, MagicMike23, Backin'the'Backs, kishi, andrewryno, CaptainCanuck, Brian MacKinney, NASCARbernet, since_98, Burls, DeDxDbacKxJroK, The so-called Beautiful and TinySarabia.
But let's give the Comment of the Night to some praise aimed at another corner of the team, one not quite as lauded as Upton. Collin Cowgill broke his ohfer-his-career, getting his first hit to lead off the eighth. And this led to Rockstarr12:
I GOT A FEVER
And the only prescription is MORE COWGILL!
"I didn't mean to hit the umpire with the dirt, but I did mean to hit that bastard in the stands." -Babe Ruth
by Rockkstarr12 on Jul 30, 2011 9:59 PM MST up reply Unrec? Delete Hide 2
Let's do this again tomorrow: take the series and move to within two of San Fran. Who's with me?
Snake Bytes: 01/30 - 10 games under .500 Vegas' verdict on the Diamondbacks is out.
Ketel Marte Ranked #8 on "MLB Now's Top 10 Second Basemen Right Now! I think we'd settle for that position in 2023...


In order to provide guidance on optimal strategies, we looked deeply at two groups of research participants: those getting the most benefits to date from the IoT and those getting the fewest benefits.
We then compared the Leaders' survey answers to those of the firms with the lowest revenue gain last year from IoT initiatives. We refer to this group as the 'IoT Followers.' They numbered 180 in total, or about the bottom quartile (23%) of the survey population on the revenue impact question. Their revenue gains were between 0% and 5%, with an average increase of 2.5%. That means Leaders' revenue increase from IoT was more than 25 times that of Followers'.


What does VOD stand for?
VOD stands for Video On Demand content. This is different from live streaming as its previously recorded audio and video. These are uploaded video files to the DaCast system that can then be used from our video player.
What does GB stand for and how does it relate to DaCast?
What embed code should I use?


Found your site whilst looking simulation racing forums.
In celebration of our new onlice store just launched this week and with Christmas coming up, we are having a sale UNTIL DECEMBER 31st 2006, some good savings to be had.
Time to start bribing the other half for a new toy for Christmas.
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