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\begin{document}
\begin{footnotesize}
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%	POSTER HEADER 
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% The header is divided into two boxes:
% The first is 75% wide and houses the title, subtitle, names, university/organization and contact information
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\includegraphics[width=50mm]{Figure/TPU_rogo.png}
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\begin{minipage}[h]{0.85\linewidth}
\Large \color{NavyBlue} \textbf{Interpolation Search for Swarm Intelligence
to Optimization Problem} \color{Black}\\ \\% Title
%\Huge\textit{Country Update}\\[2.4cm] % Subtitle
\large \textbf{山本　聖也}\hspace{20mm} % Author(s)
\normalsize 富山県立大学工学部電子・情報工学科情報基盤工学講座　\\[0.4cm] % University/organization
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%	ABSTRACT
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%\color{Navy} % Navy color for the abstract

%-----------------
{\normalsize{\bf {\ \ 要約}}}
%-----------------



Swarm Intelligence is the optimization technique
based on the action of groups, such as a bird and a
fish, and a colony of the ant. PSO which is one of the
technique is developed and applied in various studies.
However, convergence of PSO is groundless. From the
point of view, in this paper, we have proposed a new hybrid
dynamical system of Swarm Intelligence and Neural
Network dynamics for finding better optimal solution.
As the main results of this paper, we first show
how to combine PSO and neural network mechanism by
the theoretical analysis, and confirmed that proposed
system can realize interpolation search based on global
information of objective environment.
Keywords: Particle Swarm Optimization, Neural
Network Dynamics

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%	INTRODUCTION
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\color{Black} % SaddleBrown color for the introduction
%-----------------
\vspace{5mm}
{\normalsize{\bf {1\ \ はじめに}}}
%-----------------

Particle swarm optimization (PSO) is the technique of
finding the optimal solution in consideration of the past
search history from the best information (p-best) which
a solid (particle) has, and the optimal value (g-best) of
the group (swarm) formed from the solid. It is a parallel
evolutionary computation technique developed by
Kennedy [3] based on the social behavior. A standard
study on PSO, treating both the social and computational
method, is [4]. The PSO algorithm is initialized
with a population of random candidate solutions, conceptualized
as particles. Each particle is assigned a
randomized velocity and is iteratively moved through
the problem space. It is attracted towards the location
of the best fitness achieved so far by the particle itself
and by the location of the best fitness achieved so far
across the whole population.
PSO is calculated updating the position and velocity
of an each particle (see Fig. 1). The discrete time PSO
algorithm is as follows:

\begin{eqnarray}
&{}& {x^{k+1}_{d}} = {x^k_{d} + v^{k+1}_{d}}
\end{eqnarray}

where $x^k_{d}$ is the position and vk
d is the velocity of an
individual $d$ of the $k$-th iteration, $r_{1}$ and $r_{2}$ are [0,1]
random number, $c_{1}$ and $c_{2}$ are parameters. In addition,
w is a parameter to be called momentum, $x^k_{db}$ $x^k_{gb}$
shows the p-best and g-best respectively. Some stability
analysis for the discrete time PSO have been discussed
[5-7]. Recently, the development of the computer sci-

\begin{center}
\includegraphics[width=130mm]{point.png}

図$1$\ \ Move of search point.
\end{center}

ence of hardware and software is remarkable. In particular,
the evolution method is performed flourishingly
and developed and applied from the easiness of the application
with the simulation base [2]. For example, hybrid
method of PSO and Simplex method is suggested
[1]. And also, method of PSO and GA (Genetic Algorithm)
is suggested [2]. By the way, the importance of
the optimization engineering for the large-scale problem
rises more and more. Searching problems of log
and pass also become large-scale by rise of Social Network
Service (SNS). It takes time to solve these problems
with the latest computer. Therefore, we provide
the new hybrid dynamical system that it is found the
most optimal solution with the number of few steps.
From Continuous PSO algorithm, we obtain the new
dynamical system Σn and we can formulate equivalent
dynamics to a neural network from the continuous time
PSO algorithm. Then, we consider application of reinforcement
learning mechanism to realize interpolation
search of global optimum.
This paper is organized as follows. In section 2, we
briefly describe the Continuous PSO algorithm. Next,
we present the new dynamical system and derive it with
steepest descent method In section 3. Then, we provide
numerical experiments to show the effectiveness of the
presented dynamics In section 4. Finally, Section 5 concludes
the paper.

%-------------------------
\vspace{5mm}
{\normalsize{\bf {2\ \ Continuous PSO algorithm}}}

%-------------------------

Now we consider the feasible region of problem solution
and represent the continuous time PSO dynamics
by matrices as follows Σ$m$:

\begin{eqnarray}
&{}& \dot{X} = V\\
&{}& \dot{V} = -\alpha V + \beta (X_{db} - X) + \gamma (X_{gb}T - X) 
\end{eqnarray}

However, it should be noted that $X$ and $V$ are not the only system states, $X_{db}$ is also a state since it has a `memory'.
\begin{eqnarray}
&{}& \dot{X_{db}} = a(X - X_{db})[I_n + {\rm diag} [ {\rm sgn} (F(X_{gb} - F(X))]]  \nonumber \\
&{}& \\
&{}& X_{gb} = X_{db}Q_j \hspace{3mm}{\rm where} \ j = \arg {\inf_{0<i \leq n}(f(x_{db_i}))}
\end{eqnarray}
where $d$ denotes the problem dimension, the feasible region is represented by $\Omega \subseteq {\Re}^d$, $n$ denotes the number of birds, $f: \Omega \rightarrow {\Re}$ is the function to be minimized. CPSO is expected to give better solutions than standard PSO. In addition, for the sake of  compactness, the following vectors and matrices are defined as follows:
\vspace{2mm}
\begin{enumerate}
\item[$\bullet$]
$X \stackrel{\triangle}{=} [x_1 \cdots x_n] \in \Re^{d \times n}$ the position matrix
\item[$\bullet$]
$V \stackrel{\triangle}{=} [v_1 \cdots v_n] \in \Re^{d \times n}$ the velocity matrix
\item[$\bullet$]
$X_{db} \stackrel{\triangle}{=} [x_{db_1} \cdots x_{db_n}] \in \Re^{d \times n}$ the local best position matrix
\item[$\bullet$]
$X_{gb} \in {\Re}^d$ the global best position matrix
\item[$\bullet$]
$F \stackrel{\triangle}{=} [f(x_1) \cdots f(x_n)] : \Re^{d \times n} \rightarrow \Re^n$ is the stacked objective function raw vector
\item[$\bullet$]
$T$ is a row vector composed of ones
\item[$\bullet$]
$Q_i \in {\Re}^n$ is a column vector having all elements equal to zero except the $i^{th}$ element that equals one
\item[$\bullet$]
$I_n \in \Re^{n \times n}$ is the identity matrix of size $n$
\end{enumerate}

\vspace{2mm}

Let ${\rm diag}[y]$ be a diagonal matrix with the diagonal elements given by the elements of the vector $y$ and ${\rm sgn}(y)$ denotes the sigma function for $y$, defined as: ${\rm sgn}(y) = 1$ if $y > 0$, and ${\rm sgn}(y) = -1$ if $y<0$. 

Hence, assuming to be a positive constant, it is proposed to approximate  the evolution of $X$ for a minimization. Also stability analysis for the CPSO has been discussed [8]. 

The notation used above is not the standard state space notation since the state variable $X$, $V$ and $X_{db}$ are not vectors but rather matrices of the adequate dimension previously defined. This description is motivated by the simplicity and compactness it provides without loss of clarity.

\vspace{2mm}

\noindent{\bf [[Algorithm of CPSO]} 
\begin{description}
\item[Step 1] Set initial value of $X,\ V,\ X_{\mu}$ and parameters $\alpha,\ \beta,\ \gamma,\ \delta_{1}, \delta_{2}\ {\rm and}\ a$. 
\item[Step 2] Derive initial value of $X_{db},\ X_{gb},\ $. 
\item[Step 3] Calculate ${\dot V}$ and update $V$. 
\item[Step 4] Update $X$ and evaluate $X_{db},\ X_{gb}$.
\item[Step 5] If it assume to be converge then finish otherwise return {\bf Step 3}. 
\end{description}


%-------------------------
\vspace{5mm}
{\normalsize{\bf {3\ \ Proposal of hybrid PSO algorithm}}}
%-------------------------

We define the following $Z,$
\begin{eqnarray}
&{}& {Z} = \beta (X_{db} - X) + \gamma (X_{gb}T - X) 
\end{eqnarray}

and consider the following condition.
\begin{eqnarray}
&{}& {X} = X_{0} + \int_{0}^{t} V(s)ds 
\end{eqnarray}
where $X_0$ is the initial position matrix. Here we introduce
hybrid dynamical system Σ$_n$ (see Fig. 2):

\begin{eqnarray}
&{}& \dot{X} = V\\
&{}& \dot{V} = -\alpha V + Z\\
&{}& \dot{Z} = \beta(\dot{X_{db}}-\dot{X})+\gamma(\dot{X_{gb}}T-\dot{X})+\delta(\dot{X_{µ}})
\end{eqnarray}

where the dimensions of $X$, $V$ , $X_{db}$, $X_{gb}$, and $T$ are
as defined above; and the time notation is dropped for
brevity.

\begin{eqnarray}
&{}& \dot{X_{db}} = a(X-X_{db})[I_{n}+diag[sgn(F(X_{gb}-F(X)))]]\\
&{}& \dot{X_{gb}} = X_{db}Q_{j} 
\end{eqnarray}

where $\alpha,\beta,\gamma$ and $\delta$ are
 positive non zero reals, and
satisfy  $\beta+\gamma+\delta=1$.These reals are parameters which
you put weight to adjust PSO and neural network. $X_{µ}$ is the new matrix which derived from neural network
dynamics. $X_{µ}$ is defined the following dynamics [7] (see
Fig. 3) :

\begin{eqnarray}
&{}& \dot{x_{µi}} = -C\sum_{t=1}^{n}\frac{f(y_{i}(t))}{y_{i}}\\
&{}& \dot{X_{i}} = -ax_{i}(t) + z_{i}(t)\\
&{}& y_{i}(t) = (x_{i}(t)) 
\end{eqnarray}

Here we introduce the discrete formula, thus $z^k_{i}$ is vector
of $Z$ and $x^k_{µi}$
is vector of $X_{µ}$ about individual i of the $k-$
th iteration respectively. Values a and $C$ are parameters
and  is sigmoid function. Then, we adapt difference
method.
From a viewpoint of theoretical analysis, $\dot{V}$ of PSO and $\dot{x_{i}}$ of neural networ regard it as an equivalent thing. $\beta(\dot{X_{db}}-\dot{X})+\gamma(\dot{X_{gb}}T-\dot{X})$ controls the velocity of PSO

\begin{center}
\includegraphics[width=160mm]{point2.png}

図$2$\ \ Algorithm of proposed system Σn.
\end{center}

and $\delta(\dot{X_{µ}})$ controls a neural networl velocity. There-fore, $\dot{Z} = \beta(\dot{X_{db}}-\dot{X})+\gamma(\dot{X_{µ}})$ of the hybrid
proposed theoretically will have the global search
which PSO has, and the local steepest descent method
which a neural network has. Although the theoretical
algorithm for combination of PSO and a neural network
is considered by the continuous time model, a numerical
simulation is done by the dispersion model. A setup of
sampling time will change with the values of the coeffi-cient
of $\beta,\gamma,\delta.$

Thus, we calculate and obtain the $X_{µ}$.

\begin{center}
\includegraphics[width=160mm]{point3.png}

図$3$\ \ Dynamics of X
\end{center}

\noindent{\bf [Algorithm of convolution gradient method considering global information]} 
\begin{description}
\item[Step 1] Set initial value of $X,\ V,\ X_{\mu}$ and parameters $\alpha,\ \beta,\ \gamma,\ \delta_{1}, \delta_{2}\ {\rm and}\ a$. 
\item[Step 2] Derive initial value of $X_{db},\ X_{gb},\ {\rm and}\ Z$. 
\item[Step 3] Calculate ${\dot Z}$ and update $Z$. 
\item[Step 4] Calculate ${\dot V}$ and update $V$. 
\item[Step 5]Update $X$ and evaluate $X_{db},\ X_{gb},\ {\rm and}\ Z$ using the objective function $f$.
\item[Step 6] Update $X_{\mu}$.
\item[Step 7] If it assume to be converge then finish otherwise return {\bf Step 3}. 
\end{description}


%-------------------------
\vspace{5mm}
{\normalsize{\bf {4\ \ 数値実験}}}
%-------------------------


%-------------------------
\vspace{5mm}
{\normalsize{\bf {5\ \ おわりに}}}
%-------------------------

小売業を主体として進められてきた先行研究について説明し、その導出の前提について考察し、限界を示した。次に、自動車産業の特徴である内示生産システムについて詳述し、内示情報が発注元から提示されるブレを含む参考値であることを示した。また、内示生産システムを基本としたときの発注業務フローを示した。そのうえで、週次内示を使った場合と月次内示を使った場合のそれぞれについてブルウィップ効果の評価式を提案した。
週次内示を用いた場合には、ブルウィップ効果は1よりおおきくなる。月次内示をもちいた場合には、週次内示と確定注文のブレ、月次内示と週次内示のブレ、リードタイム増によるブレ等の要因がブルウィップ効果を大きくしている。
データ解析から内示情報のブレの特性を示し、一定の条件下でブルウィップ効果を評価した。月次内示をもちいた場合は、2.08から5.13以上である。
今回は、内示配信方法をもとに、発注業務フローを明確にし、これをもとに検討のフレームワークを示し、需要のブレを主要因とする基本的なブルウィップ効果の定量化を行った。


%-----------------
\vspace{5mm}
{\normalsize{\bf {参考文献}}}
%-----------------

\vspace{-5mm}

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\end{thebibliography}

\end{multicols}
\end{footnotesize}
\end{document}