\documentclass[paper=a4, fontsize=11pt]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage{fourier}
\usepackage[english]{babel}
\usepackage[protrusion=true,expansion=true]{microtype}
\usepackage{amsmath,amsfonts,amsthm}
\usepackage[pdftex]{graphicx}
\usepackage{url}
\usepackage{sectsty}
\usepackage{rotating}
\allsectionsfont{\centering \normalfont\scshape}

\usepackage{fancyhdr}
\pagestyle{fancyplain}
\fancyhead{}
\fancyfoot[L]{}
\fancyfoot[C]{}
\fancyfoot[R]{\thepage}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
\setlength{\headheight}{13.6pt}
\numberwithin{equation}{section}
\numberwithin{figure}{section}
\numberwithin{table}{section}
\newcommand{\horrule}[1]{\rule{\linewidth}{#1}}

\title{
		%\vspace{-1in} 	
		\usefont{OT1}{bch}{b}{n}
		\normalfont \normalsize \textsc{Central Washington University of the Computer Science Department} \\ [25pt]
		\horrule{0.5pt} \\[0.4cm]
		\huge Project 4 \\
		\horrule{2pt} \\[0.5cm]
}

\author{\normalsize Mitchell Hansen \\[-6pt]}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\maketitle

\section{Introduction}
For this lab we again took our 15 optimization functions and ran them through
3 new methods of determining the global minimum. The functions being:
The Self Organizing Migrating Algorithm (SOMA) which uses an evolutional approach
, The Firefly Algorithm (FA) which uses an evolutional swarm approach
similar to Particle Swarm, and the Harmony Search Algorithm (HS) which uses
another evolutional style approach.

\section{Methods}
For each of the 3 search methods and all 15 of the search functions we ran
tests for a set amount of iterations using the python scripts we wrote
for the previous lab. These results were then written out to a file from which
we calculated the min, max, range, etc.

We also slightly modified the way FA calculated the step when changin it's direction.
Instead of using a random distribution when calculating the FA random step
we used a suggested regular distribution that we found when researching the
problem online.

\section{Analysis}
This lab produced both surprising and disappointing results when evaluating the search functions.
The most surprising results are that of SOMA, where it regularly equaled or beat the most accurate
search function so far which has been Differential Evolution (DE). Of the 15 functions tested, SOMA
had better results than DE for 6 of the functions. The rest of the 9 functions tested were very close
to the results found by the DE search function.

The disappointing results produced by this lab lay in the other two functions tested. HS and FA produced
very similar results overall, most possibly because they are not all that much better than particle
swarm of which they are in the same class. Between the two search functions, there was very seldom
a difference of more than 20 to 30 percent in their fitnesses. We think it's best to compare these
two functions to their cousin Particle Swarm (PSO) as it is the closest related method.

In relation to PSO the functions did very similarly for the later functions (>4). The lower functions (<4)
all produced results which were much more inaccurate than traditional particle swarm. In regards to the
later functions, HS was only able to beat a PSO search method under function 6 with 11.28 determined as
the minimum, with PSO returning 12.15. FA wasn't able to beat PSO in function 6 but it did come close
with 13.32. Another notable point to look at is function 15 where HS was able to find the minimum at
-18.70, but FA was unable to capture the minimum with 16.40.


\section{Conclusion}

The conclusion for this lab is a pessimistic one, SOMA was a great success and a valuable
addition to our growing library of search functions. But are overshadowed by the poor performance
not only in the accuracy of answers, but also the run times of FA and HS. PSO was able to
outperform HS and FA in almost every aspect, and that is even telling as PSO performs
poorly when compared to other search functions we have used for some specific functions.
Walking away from this lab, I believe that we will add SOMA to our list of highly accurate
and performance search methods, while leaving FA and HS for problem sets which they are most
suited to.

Although we scrutinized our code to an appropriate degree we can't rule out the poor performance
of the FA and HS algorithms being caused by incorrect implementations. We're reasonably
sure that this is not the case though as the comparable performance to PSO was expected.
Additionally SOMA was not entirely without faults either. Function 10 for example shows an almost 250
percent increase in the minimum value over it's DE an PSO counterparts. This is worrisome as
all of the other results produced by SOMA were within 20 to 30 percent of PSO and DE.


\begin{figure}
\section{Results}
\caption{Computation comparison of SOMA, HS and FA}
\hskip+4.0cm
\rotatebox{90.0}{
\scalebox{0.7}{
    \small \centering

    \label{Tab1d}
    \begin{tabular}{c|lllll|lllll|lllll}
    \noalign{\smallskip}\hline\noalign{\smallskip}
    Problem & \multicolumn{5}{c}{SOMA}
            & \multicolumn{5}{|c|}{HS}
            &  \multicolumn{5}{c}{FA} \\
    \noalign{\smallskip}\hline\noalign{\smallskip}



         & Avg       & Median     & Range       & SD        & T(s)      & Avg               & Median            & Range         & SD            & T(s)      & Avg            & Median            & Range             & SD            & T(s)     \\ \noalign{\smallskip}\hline\noalign{\smallskip}
$f_1$    & -7299.02  & -7386.83   & 1503.58     & 452.97    & 0.08      & -1736.64          & -1762.09          & 901.78        & 340.48        & 4.73      & -2078.10       & -2121.93          & 1297.24           & 956.34        & 2.00     \\
$f_2$    & 73.06     & 33.94      & 332.07      & 100.36    & 0.05      & 38678.19          & 39044.00          & 14740.90      & 5545.68       & 3.63      & 39611.35       & 41038.00          & 15876.60          & 8095.76       & 0.87     \\
$f_3$    & 149.76    & 119.57     & 338.84      & 99.33     & 0.10      & 15223266000.00    & 15499350000.00    & 9746940000.00 & 3077498860.01 & 3.87      & 13940456000.00 & 13970750000.00    & 11865890000.00    & 3765178464.71 & 1.06     \\
$f_4$    & -7758.64  & -7737.15   & 362.37  	& 131.58    & 0.20      & 132487.90         & 138380.00         & 75080.00      & 22337.69      & 4.71      & 135166.40      & 136884.00         & 22462.00          & 30028.16      & 2.09     \\
$f_5$    & 45.26     & 35.50      & 124.00      & 36.68     & 0.07      & 249.51            & 249.76            & 66.18         & 22.99         & 4.75      & 219.01         & 229.32            & 130.89            & 46.52         & 2.18     \\
$f_6$    & 13.70     & 13.84      & 2.91        & 0.99      & 0.00      & 11.28             & 11.24             & 0.82          & 0.24          & 5.90      & 13.32          & 13.25             & 1.35              & 3.11          & 2.89     \\
$f_7$    & 36.00     & 34.30      & 24.57       & 8.03      & 0.04      & 34.10             & 34.92             & 5.20          & 1.82          & 6.54      & 45.15          & 44.93             & 8.66              & 9.88          & 4.59     \\
$f_8$    & 14.49     & 15.01      & 72.42       & 28.43     & 0.10      & 279.82            & 282.06            & 57.12         & 17.79         & 5.95      & 273.29         & 278.48            & 37.49             & 62.42         & 3.66     \\
$f_9$    & 141.72    & 212.47     & 376.52      & 154.36    & 0.09      & 305.15            & 306.60            & 19.46         & 6.85          & 7.10      & 342.88         & 354.12            & 71.60             & 74.94         & 4.89     \\
$f_{10}$ & -13566.43 & -14173.65  & 3948.20 	& 1297.42   & 0.55      & -3194.74          & -2846.66          & 2290.95       & 832.19        & 6.07      & -3332.62       & -3378.12          & 2053.65           & 1555.81       & 3.43     \\
$f_{11}$ & -8428.54  & -8806.80   & 3527.50 	& 1076.29   & 0.58      & -2037.88          & -1976.69          & 1404.16       & 421.07        & 8.39      & -2009.29       & -1998.01          & 1467.36           & 1004.39       & 6.10     \\
$f_{12}$ & 8.91      & 8.88       & 0.87        & 0.27      & 0.01      & 7.78              & 7.74              & 0.59          & 0.19          & 5.76      & 8.71           & 8.77              & 0.44              & 2.14          & 3.10     \\
$f_{13}$ & -2.36     & -2.38      & 2.90        & 0.81      & 0.01      & -5.46             & -5.55             & 1.53          & 0.48          & 7.33      & -2.53          & -2.52             & 2.33              & 1.43          & 4.73     \\
$f_{14}$ & -7.39     & -7.74      & 8.78        & 3.13      & 0.01      & -12.86            & -12.81            & 2.15          & 0.67          & 5.41      & -7.47          & -7.21             & 6.35              & 4.04          & 3.20     \\
$f_{15}$ & -17.46    & -18.45     & 6.22        & 2.14      & 0.03      & -18.70            & -18.70            & 0.00          & 0.00          & 11.09     & -16.40         & -17.37            & 6.14              & 6.23          & 9.65     \\ \noalign{\smallskip}\hline\noalign{\smallskip}
     & & & & & & & & & & & & & & &       \\
\noalign{\smallskip}\hline\noalign{\smallskip} \multicolumn{16}{l}{\tiny $^1$ ThinkPad, 3.4GHz Intel Core i7 (3rd gen), 16 GB RAM}

\end{tabular}
}}

\end{figure}


\newpage

\begin{figure}
\section{Previous Results}
\caption{Computation comparison of DE, GA and PSO}
\hskip+4.0cm
\rotatebox{90.0}{
\scalebox{0.7}{
    \small \centering

    \label{Tab1d}
    \begin{tabular}{c|lllll|lllll|lllll}
    \noalign{\smallskip}\hline\noalign{\smallskip}
    Problem & \multicolumn{5}{c}{DE}& \multicolumn{5}{|c|}{GA}
            &  \multicolumn{5}{c}{PSO} \\
    \noalign{\smallskip}\hline\noalign{\smallskip}




         & Avg      & Median    & Range     & SD        & T(s)      & Avg           & Median        & Range         & SD            & T(s)      & Avg       & Median        & Range         & SD            & T(s)     \\ \noalign{\smallskip}\hline\noalign{\smallskip}
$f_1$    & -6112.33 & -6084.59  & 114.26    & 47.83     & 1.14      & -3276.12      & -3292.95      & 943.02        & 245.68        & 2.69      & -2871.98  & -2904.39      & 1194.77       & 322.06        & 0.12     \\
$f_2$    & 129.53   & 25.00     & 900.00	& 251.52    & 0.53      & 23185.53      & 22853.00      & 10310.00      & 3148.43       & 0.72      & 0.17      & 0.15          & 0.25          & 0.08          & 0.09     \\
$f_3$    & 26105.67 & 10019.00  & 168100.00 & 43662.88  & 0.78      & 5291234666.67 & 5017400000.00 & 5739020000.00 & 1539343402.74 & 0.68      & 421.98    & 200.19        & 1657.68       & 497.31        & 0.10     \\
$f_4$    & -7600.00 & -7960.00  & 2560.00	& 728.99    & 1.00      & 79752.00      & 81520.00      & 23240.00      & 8507.40       & 2.12      & -5206.62  & -5324.98      & 3479.78       & 1178.83       & 0.13     \\
$f_5$    & 0.00     & 0.00      & 0.00	    & 0.00      & 1.08      & 145.86        & 150.55        & 51.89         & 17.68         & 2.31      & 9.17      & 8.93          & 5.88          & 1.95          & 0.13     \\
$f_6$    & 12.38    & 12.71     & 2.19	    & 0.60      & 1.46      & 12.04         & 11.97         & 0.67          & 0.22          & 2.52      & 12.15     & 12.18         & 1.25          & 0.33          & 0.14     \\
$f_7$    & 19.06    & 19.01     & 0.62	    & 0.16      & 1.67      & 36.69         & 36.60         & 5.76          & 1.54          & 4.20      & 20.55     & 20.45         & 2.63          & 0.68          & 0.18     \\
$f_8$    & 58.74    & 58.73     & 4.74	    & 1.54      & 1.60      & 212.86        & 213.95        & 41.20         & 11.06         & 3.41      & -9.92     & -11.64        & 35.51         & 9.72          & 0.10     \\
$f_9$    & -83.30   & -80.69    & 21.87	    & 6.99      & 2.09      & 276.38        & 276.83        & 14.65         & 4.35          & 4.10      & 251.53    & 288.37        & 173.05        & 64.83         & 0.14     \\
$f_{10}$ & -4959.12 & -4579.12  & 2896.23	& 966.10    & 3.02      & -4778.37      & -4822.17      & 978.82        & 327.79        & 4.72      & -4107.05  & -3830.50      & 2663.98       & 711.61        & 0.13     \\
$f_{11}$ & -8478.48 & -8821.20  & 5161.40	& 1330.20   & 3.56      & -3188.30      & -3181.83      & 1334.30       & 339.30        & 8.34      & -2899.33  & -2888.72      & 901.67        & 227.81        & 0.21     \\
$f_{12}$ & 0.00     & 0.00      & 0.00	    & 0.00      & 1.48      & 8.00          & 8.01          & 0.69          & 0.17          & 2.70      & 7.02      & 7.08          & 1.30          & 0.37          & 0.15     \\
$f_{13}$ & -4.28    & -4.22     & 2.71	    & 0.83      & 3.06      & -4.27         & -4.22         & 2.30          & 0.57          & 5.54      & -10.39    & -9.86         & 4.92          & 1.50          & 0.14     \\
$f_{14}$ & -18.99   & -19.00    & 0.04	    & 0.01      & 1.47      & -10.88        & -10.53        & 3.70          & 1.00          & 3.65      & -16.07    & -16.15        & 5.22          & 1.59          & 0.14     \\
$f_{15}$ & -21.91   & -23.03    & 8.39      & 2.95      & 6.05      & -14.64        & -14.64        & 0.00          & 0.00          & 12.55     & -18.70    & -18.70        & 0.00          & 0.00          & 0.27     \\ \noalign{\smallskip}\hline\noalign{\smallskip}
     & & & & & & & & & & & & & & &       \\
\noalign{\smallskip}\hline\noalign{\smallskip} \multicolumn{16}{l}{\tiny $^1$ ThinkPad, 3.4GHz Intel Core i7 (3rd gen), 16 GB RAM}

\end{tabular}
}}

\end{figure}



\end{document}

