\documentclass[11pt,twoside]{article}\makeatletter \IfFileExists{xcolor.sty}% {\RequirePackage{xcolor}}% {\RequirePackage{color}} \usepackage{colortbl} \usepackage{wrapfig} \usepackage{ifxetex} \ifxetex \usepackage{fontspec} \usepackage{xunicode} \catcode`⃥=\active \def⃥{\textbackslash} \catcode`❴=\active \def❴{\{} \catcode`❵=\active \def❵{\}} \def\textJapanese{\fontspec{Noto Sans CJK JP}} \def\textChinese{\fontspec{Noto Sans CJK SC}} \def\textKorean{\fontspec{Noto Sans CJK KR}} \setmonofont{DejaVu Sans Mono} \else \IfFileExists{utf8x.def}% {\usepackage[utf8x]{inputenc} \PrerenderUnicode{–} }% {\usepackage[utf8]{inputenc}} \usepackage[english]{babel} \usepackage[T1]{fontenc} \usepackage{float} \usepackage[]{ucs} \uc@dclc{8421}{default}{\textbackslash } \uc@dclc{10100}{default}{\{} \uc@dclc{10101}{default}{\}} \uc@dclc{8491}{default}{\AA{}} \uc@dclc{8239}{default}{\,} \uc@dclc{20154}{default}{ } \uc@dclc{10148}{default}{>} \def\textschwa{\rotatebox{-90}{e}} \def\textJapanese{} \def\textChinese{} \IfFileExists{tipa.sty}{\usepackage{tipa}}{} \fi \def\exampleFont{\ttfamily\small} \DeclareTextSymbol{\textpi}{OML}{25} \usepackage{relsize} \RequirePackage{array} \def\@testpach{\@chclass \ifnum \@lastchclass=6 \@ne \@chnum \@ne \else \ifnum \@lastchclass=7 5 \else \ifnum \@lastchclass=8 \tw@ \else \ifnum \@lastchclass=9 \thr@@ \else \z@ \ifnum \@lastchclass = 10 \else \edef\@nextchar{\expandafter\string\@nextchar}% \@chnum \if \@nextchar c\z@ \else \if \@nextchar l\@ne \else \if \@nextchar r\tw@ \else \z@ \@chclass \if\@nextchar |\@ne \else \if \@nextchar !6 \else \if \@nextchar @7 \else \if \@nextchar (8 \else \if \@nextchar )9 \else 10 \@chnum \if \@nextchar m\thr@@\else \if \@nextchar p4 \else \if \@nextchar b5 \else \z@ \@chclass \z@ \@preamerr \z@ \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi} \gdef\arraybackslash{\let\\=\@arraycr} \def\@textsubscript#1{{\m@th\ensuremath{_{\mbox{\fontsize\sf@size\z@#1}}}}} \def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}} \def\abbr{} \def\corr{} \def\expan{} \def\gap{} \def\orig{} \def\reg{} \def\ref{} \def\sic{} \def\persName{}\def\name{} \def\placeName{} \def\orgName{} \def\textcal#1{{\fontspec{Lucida Calligraphy}#1}} \def\textgothic#1{{\fontspec{Lucida Blackletter}#1}} \def\textlarge#1{{\large #1}} \def\textoverbar#1{\ensuremath{\overline{#1}}} \def\textquoted#1{‘#1’} \def\textsmall#1{{\small #1}} \def\textsubscript#1{\@textsubscript{\selectfont#1}} \def\textxi{\ensuremath{\xi}} \def\titlem{\itshape} \newenvironment{biblfree}{}{\ifvmode\par\fi } \newenvironment{bibl}{}{} \newenvironment{byline}{\vskip6pt\itshape\fontsize{16pt}{18pt}\selectfont}{\par } \newenvironment{citbibl}{}{\ifvmode\par\fi } \newenvironment{docAuthor}{\ifvmode\vskip4pt\fontsize{16pt}{18pt}\selectfont\fi\itshape}{\ifvmode\par\fi } \newenvironment{docDate}{}{\ifvmode\par\fi } \newenvironment{docImprint}{\vskip 6pt}{\ifvmode\par\fi } \newenvironment{docTitle}{\vskip6pt\bfseries\fontsize{22pt}{25pt}\selectfont}{\par } \newenvironment{msHead}{\vskip 6pt}{\par} \newenvironment{msItem}{\vskip 6pt}{\par} \newenvironment{rubric}{}{} \newenvironment{titlePart}{}{\par } \newcolumntype{L}[1]{){\raggedright\arraybackslash}p{#1}} \newcolumntype{C}[1]{){\centering\arraybackslash}p{#1}} \newcolumntype{R}[1]{){\raggedleft\arraybackslash}p{#1}} \newcolumntype{P}[1]{){\arraybackslash}p{#1}} \newcolumntype{B}[1]{){\arraybackslash}b{#1}} \newcolumntype{M}[1]{){\arraybackslash}m{#1}} \definecolor{label}{gray}{0.75} \def\unusedattribute#1{\sout{\textcolor{label}{#1}}} \DeclareRobustCommand*{\xref}{\hyper@normalise\xref@} \def\xref@#1#2{\hyper@linkurl{#2}{#1}} \begingroup \catcode`\_=\active \gdef_#1{\ensuremath{\sb{\mathrm{#1}}}} \endgroup \mathcode`\_=\string"8000 \catcode`\_=12\relax \usepackage[a4paper,twoside,lmargin=1in,rmargin=1in,tmargin=1in,bmargin=1in,marginparwidth=0.75in]{geometry} \usepackage{framed} \definecolor{shadecolor}{gray}{0.95} \usepackage{longtable} \usepackage[normalem]{ulem} \usepackage{fancyvrb} \usepackage{fancyhdr} \usepackage{graphicx} \usepackage{marginnote} \renewcommand{\@cite}[1]{#1} \renewcommand*{\marginfont}{\itshape\footnotesize} \def\Gin@extensions{.pdf,.png,.jpg,.mps,.tif} \pagestyle{fancy} \usepackage[pdftitle={Dynamic Programming and Taguchi Method Optimization of Water-Treatment-Plant Design}, pdfauthor={}]{hyperref} \hyperbaseurl{} \paperwidth210mm \paperheight297mm \def\@pnumwidth{1.55em} \def\@tocrmarg {2.55em} \def\@dotsep{4.5} \setcounter{tocdepth}{3} \clubpenalty=8000 \emergencystretch 3em \hbadness=4000 \hyphenpenalty=400 \pretolerance=750 \tolerance=2000 \vbadness=4000 \widowpenalty=10000 \renewcommand\section{\@startsection {section}{1}{\z@}% {-1.75ex \@plus -0.5ex \@minus -.2ex}% {0.5ex \@plus .2ex}% {\reset@font\Large\bfseries}} \renewcommand\subsection{\@startsection{subsection}{2}{\z@}% {-1.75ex\@plus -0.5ex \@minus- .2ex}% {0.5ex \@plus .2ex}% {\reset@font\Large}} \renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% {-1.5ex\@plus -0.35ex \@minus -.2ex}% {0.5ex \@plus .2ex}% {\reset@font\large}} \renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% {-1ex \@plus-0.35ex \@minus -0.2ex}% {0.5ex \@plus .2ex}% {\reset@font\normalsize}} \renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}% {1.5ex \@plus1ex \@minus .2ex}% {-1em}% {\reset@font\normalsize\bfseries}} \def\l@section#1#2{\addpenalty{\@secpenalty} \addvspace{1.0em plus 1pt} \@tempdima 1.5em \begingroup \parindent \z@ \rightskip \@pnumwidth \parfillskip -\@pnumwidth \bfseries \leavevmode #1\hfil \hbox to\@pnumwidth{\hss #2}\par \endgroup} \def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}} \def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}} \def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}} \def\l@subparagraph{\@dottedtocline{5}{10em}{5em}} \@ifundefined{c@section}{\newcounter{section}}{} \@ifundefined{c@chapter}{\newcounter{chapter}}{} \newif\if@mainmatter \@mainmattertrue \def\chaptername{Chapter} \def\frontmatter{% \pagenumbering{roman} \def\thechapter{\@roman\c@chapter} \def\theHchapter{\roman{chapter}} \def\thesection{\@roman\c@section} \def\theHsection{\roman{section}} \def\@chapapp{}% } \def\mainmatter{% \cleardoublepage \def\thechapter{\@arabic\c@chapter} \setcounter{chapter}{0} \setcounter{section}{0} \pagenumbering{arabic} \setcounter{secnumdepth}{6} \def\@chapapp{\chaptername}% \def\theHchapter{\arabic{chapter}} \def\thesection{\@arabic\c@section} \def\theHsection{\arabic{section}} } \def\backmatter{% \cleardoublepage \setcounter{chapter}{0} \setcounter{section}{0} \setcounter{secnumdepth}{2} \def\@chapapp{\appendixname}% \def\thechapter{\@Alph\c@chapter} \def\theHchapter{\Alph{chapter}} \appendix } \newenvironment{bibitemlist}[1]{% \list{\@biblabel{\@arabic\c@enumiv}}% {\settowidth\labelwidth{\@biblabel{#1}}% \leftmargin\labelwidth \advance\leftmargin\labelsep \@openbib@code \usecounter{enumiv}% \let\p@enumiv\@empty \renewcommand\theenumiv{\@arabic\c@enumiv}% }% \sloppy \clubpenalty4000 \@clubpenalty \clubpenalty \widowpenalty4000% \sfcode`\.\@m}% {\def\@noitemerr {\@latex@warning{Empty `bibitemlist' environment}}% \endlist} \def\tableofcontents{\section*{\contentsname}\@starttoc{toc}} \parskip0pt \parindent1em \def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}} \newenvironment{reflist}{% \begin{raggedright}\begin{list}{} {% \setlength{\topsep}{0pt}% \setlength{\rightmargin}{0.25in}% \setlength{\itemsep}{0pt}% \setlength{\itemindent}{0pt}% \setlength{\parskip}{0pt}% \setlength{\parsep}{2pt}% \def\makelabel##1{\itshape ##1}}% } {\end{list}\end{raggedright}} \newenvironment{sansreflist}{% \begin{raggedright}\begin{list}{} {% \setlength{\topsep}{0pt}% \setlength{\rightmargin}{0.25in}% \setlength{\itemindent}{0pt}% \setlength{\parskip}{0pt}% \setlength{\itemsep}{0pt}% \setlength{\parsep}{2pt}% \def\makelabel##1{\upshape ##1}}% } {\end{list}\end{raggedright}} \newenvironment{specHead}[2]% {\vspace{20pt}\hrule\vspace{10pt}% \phantomsection\label{#1}\markright{#2}% \pdfbookmark[2]{#2}{#1}% \hspace{-0.75in}{\bfseries\fontsize{16pt}{18pt}\selectfont#2}% }{} \def\TheFullDate{2011-06-04 (revised: 04 June 2011)} \def\TheID{\makeatother } \def\TheDate{2011-06-04} \title{Dynamic Programming and Taguchi Method Optimization of Water-Treatment-Plant Design} \author{}\makeatletter \makeatletter \newcommand*{\cleartoleftpage}{% \clearpage \if@twoside \ifodd\c@page \hbox{}\newpage \if@twocolumn \hbox{}\newpage \fi \fi \fi } \makeatother \makeatletter \thispagestyle{empty} \markright{\@title}\markboth{\@title}{\@author} \renewcommand\small{\@setfontsize\small{9pt}{11pt}\abovedisplayskip 8.5\p@ plus3\p@ minus4\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus2\p@ \belowdisplayshortskip 4\p@ plus2\p@ minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 2\p@ plus1\p@ minus1\p@ \parsep 2\p@ plus\p@ minus\p@ \itemsep 1pt} } \makeatother \fvset{frame=single,numberblanklines=false,xleftmargin=5mm,xrightmargin=5mm} \fancyhf{} \setlength{\headheight}{14pt} \fancyhead[LE]{\bfseries\leftmark} \fancyhead[RO]{\bfseries\rightmark} \fancyfoot[RO]{} \fancyfoot[CO]{\thepage} \fancyfoot[LO]{\TheID} \fancyfoot[LE]{} \fancyfoot[CE]{\thepage} \fancyfoot[RE]{\TheID} \hypersetup{citebordercolor=0.75 0.75 0.75,linkbordercolor=0.75 0.75 0.75,urlbordercolor=0.75 0.75 0.75,bookmarksnumbered=true} \fancypagestyle{plain}{\fancyhead{}\renewcommand{\headrulewidth}{0pt}} \date{} \usepackage{authblk} \providecommand{\keywords}[1] { \footnotesize \textbf{\textit{Index terms---}} #1 } \usepackage{graphicx,xcolor} \definecolor{GJBlue}{HTML}{273B81} \definecolor{GJLightBlue}{HTML}{0A9DD9} \definecolor{GJMediumGrey}{HTML}{6D6E70} \definecolor{GJLightGrey}{HTML}{929497} \renewenvironment{abstract}{% \setlength{\parindent}{0pt}\raggedright \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt \textcolor{GJBlue}{\large\bfseries\abstractname\space} }{% \vskip8pt \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt } \usepackage[absolute,overlay]{textpos} \makeatother \usepackage{lineno} \linenumbers \begin{document} \author[1]{Dr. Khezri-S.M} \affil[1]{ Islamic Azad University, Tehran, Iran} \renewcommand\Authands{ and } \date{\small \em Received: 26 April 2011 Accepted: 20 May 2011 Published: 4 June 2011} \maketitle \begin{abstract} System and process optimization are both necessary for overall system design in system optimization models, the objective is to determine the optimal system configuration while assuming the design and operating parameters of each individual process. In process optimization models the objective is to find the design and operating parameters of individual processes. The major disadvantages of GP and NLP algorithms are that global optima is not assured and they can not be directly used for system optimization because of the presence of discrete decision variables and required high computer storage [Dharmappa 1994] [Desmond F. Lawler. 2005] Using DP solely for large problems requires very high computer time for Data Processing [Mhiaisalkar 1993]. The model of this research is DP model that employs Taguchi Experimental Design Method, whereupon the time of model analysis has reduced considerably. . Case study showed the capability of model in savings of roughly 9.5% in capital and annualized costs, compared to the conventional design besides the software has capability to performing sensitivity analysis and showing interactions between various Decision variables. \end{abstract} \keywords{Optimization of water Treatment - Plants, Process Optimization - Taguchi Design of Experiments.} \begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) \textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark} \end{textblock*} \begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) \textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}} \end{textblock*} \let\tabcellsep& \section[{INTRODUCTION}]{INTRODUCTION}\par he water Industry is one of the principal industries in every country. The total expenditure for water supply, including operation and maintenance, accounts to nearly 6 billion dollars per year in the U.S alone, of this approximately 15\% is spent on actual water treatment \hyperref[b2]{(Wiesner, 1985)}. This massive investment requirement calls for more economical system design to allow for an efficient use of public funds. This objective can be achieved using "Optimization". Because in many cases pilot scale studies are conducted prior to full-scale plant construction, the use of an optimization model can greatly reduce the time and money spent on the pilot scale study.\par A water treatment system is a combination of several unit processes. Design of such a system is a difficult task and the least cost design is most difficult of all. The principal unit process in a conventional watertreatment systemincludes coagulation-flocculation, sedimentation, filtration and disinfection. The performance of each treatment unit affects the efficacy of the subsequent units. Ideally, therefore, the design decisions should be made with regard to the interaction between various unit operations. So for very few works have been done on the optimization of a water treatment system. \hyperref[b4]{Letterman \& Iyer (1977)} analyzed the effects of selected process decision variables on overall system cost and performance. The authors used simplified process models (mostly empirical relationships). One comprehensive work is that of {\ref Wiesner et. al(1978)}which is an important step towards the optimal design of water treatment system. The authors have provided a number of optimization models for integral analysis and economic optimization of the components of a water treatment plant. {\ref Mhiasalkar(1993)} has developed a mathematical model incorporating the Performance relationships and cost functions for the component units of conventional water treatment system, and an algorithm using dynamic programming has described by them for functional and minimal cost design of the system. The last and one of the comprehensive models in this ground belongs to \hyperref[b0]{Dharmappa et. al (1994)}. The authors believe the incorporation of particle size distribution (PSD) is necessary for optimal process design and selection. In this context this works includes not only all three levels of system design but also process design and selection using PSD and the Algorithm they have used was a GP and NLP programming.\par The major disadvantages of GP and NLP algorithms are that, global optima is not assured and they can not be directly used for system optimization because of the presence of discrete decision variables and requirement high computer storage. Using Dynamic Programming solely for large problems requires very high computer storage. Accordingly, this paper addresses it self to the development of a model for minimal cost and Energy design of a conventional water treatment plant using dynamic programming and Taguchi Design Of Experiments methods optimization model. The scope of this research was restricted to the economic optimization of conventional water treatment system for turbidity removal comprising four water treatment processes, which are listed in table 1.\par The serial and interactive nature of the various unit processes favors application of Dynamic Programming for minimal cost design. Dynamic Programming is a mathematical technique often useful for making a sequence of interrelated decisions. It provides a systematic procedure for determining the combination of decisions that maximizes overall effectiveness {\ref [Hillier 1967}]. Deterministic dynamic programming can be described diagrammatically, as shown in Fig. {\ref 1} Fig. {\ref 1}: The basics structure for deterministic dynamic programming.\par Thus at stage n the process will be in some state S n . Making policy decision X n then moves the process to some state S n+1 at stage (n+1). From that point onward the objective function value for the optimal policy has been previously calculated to be f* n+1 (S n+1 ). Conventional water Treatment systems can be considered as a multistage process, with the stages represented by various unit processes. The states represented by the levels of water quality parameters such as turbidity and suspended solids . The decision of unit.\par variables are design parameters depending on the type Traditionally, measuring nonspecific parameters like turbidity units (Tu) suspended solids (SS); etc assesses the performance of water treatment plants. Though their parameters describe the Important aspects regarding the water quality of the effluent form the treatment processes (AWWA: water 1990). Insofar as turbidity is used as a measure of the efficiency of the removal of particulate matter throughout the purification process of water, low turbidity in the finished water is an indication of effectiveness of unit processes (Guidelines 1984). Because mathematical models of water treatment describe the behavior of systems in relation to the removal of mass concentration of suspended solids. And we can in some ways correlate turbidity measurements with suspended solids data, suspended solids has been used as a state parameter to indicate the effectiveness of the system at various stages of treatment. Fig. \hyperref[fig_0]{2} Show the sequence of stages and the stage return for each input-decision combination. \section[{II.}]{II.} \section[{TAGUCHI METHOD}]{TAGUCHI METHOD}\par Taguchi approach to parameter design provides the design engineer with a systematic and efficient method for determining near optimum design parameters for performance and cost \hyperref[b7]{(Kackar, 1985} {\ref (Kackar, : phadke, 1989;;}\hyperref[b9]{Taguchi, 1986)}. In this research we have used orthogonal arrays from design of Experiments theory to study a large number of variables with a Small number of experiments. Whereupon the time of DP model analysis reduced considerably. 2011 Functional relationships for each unit process, linking input and output states parameters with a characteristic loading parameter, have been established based on the mechanics of unit process. These relationships are largely empirical models. Table \hyperref[tab_1]{2} shows these relationships and key assumptions for each of the process models.August R4=f(Q,T r , G r ) R3=f(Q, T s ,G s ) R2=f(Sor, U n , SS 3 , SS 2 ) R 1 =f(Fr, \section[{III.}]{III.} \section[{COST FUNCTIONS}]{COST FUNCTIONS}\par Since the present study seeks to incorporate multiple design criteria in the optimization, there are three design criteria to be evaluated from the cost functions:1. Annualized Capital Cost (ACC) 2. Operation \& Maintenance cost (O \& M) 3. \section[{Energy requirement (ER)}]{Energy requirement (ER)}\par There is a substantial literature on cost functions for water treatment plant component \hyperref[b10]{(Hinomoto, 1977;}\hyperref[b11]{Clark, 1982)}. The annualized capital and operation and maintenance cost functions are most taken from \hyperref[b11]{Clark (1982)}. Some changes have been made by Dharmappa (1994) so as to make them more generals. Capital cost functions as presented by \hyperref[b11]{Clark (1982)} were based on the annualization factor 0.102 (interest 8\% and amortization 20 years). There were modified to incorporate a general annualization factor, of, which can be calculated by (assuming zero salvation value:\par Where a f = annualization factor; i= interest rate, Fraction; and n= amortization period, years. The relationships for energy requirement are developed from \hyperref[b13]{Gumerman et al. (1979)}. Cost functions for unit processes (extracted and compiled from \hyperref[b11]{Clark, 1982}\hyperref[b0]{, with Dharmappa (1994)} \section[{MODEL OPTIMIZATION}]{MODEL OPTIMIZATION}\par There are a number of designs that satisfy water quality standards. The objective, therefore, is to minimize the system cost satisfying all constrains. The bounds used in the optimizations are given in {\ref table} DHR PR A OM a CCI A ACC f f f = = 4 B Filter surface wash 4 Opt .? [R i ] i=1 V. \section[{RECURSIVE EQUATIONS}]{RECURSIVE EQUATIONS}\par For deriving Recursive equations it is necessary the Return functions for concrete, steel, filter sand and appurtenances are also necessary. Data on discount rate, and chemical costs are required for economic evaluation. The algorithm involves four stages in DP programming, and the optimal function of the previous stage is implemented in the next stage. In each stage in addition to the previous optimal function, the return function concerning to the current stage is also used to optimize the existing stage. This procedure will be followed until the last stage. The solution of this system is computed by backward form. With this parametric solution of stages, we can derive the optimal objective function in a parametric form. This parametric form includes all design parameters of the water treatment system. This objective function could be optimized by appropriate software based on the Taguchi Design of Experiments again could be calculated.+ = Flocculation G r G r Q A Q A o T Q T Q R r \section[{VII.}]{VII.} \section[{CASE STUDY}]{CASE STUDY}\par The application of the optimization algorithm is illustrated for a 6250-m3/hr conventional water treatment plant. The optimal results of this model have been compared with EPA traditional water Treatment plant. Design criteria and cost calculation of EPA conventional water treatment plant exists in EPA Documents \hyperref[b13]{Gumerman et al. (1979)}. The optimal design results in saving of 9.5\% in capital and Annualized cost. The comparison is presented in table 8. \section[{SENSITIVITY ANALYSIS}]{SENSITIVITY ANALYSIS}\par One particularly important feature of this model is the ease with which a sensitivity analysis can be carried out. This technique analyzes the sensitivity of the optimal solution to changes in various decision variables without re-solving the problem for each new value. It is highly unlikely that cost data of sufficient accuracy will ever be available \hyperref[b15]{(Walski 1991)}. Therefore, variation in total Annualized cost was also studied. The results of this analysis have been presented in table {\ref 9}. \section[{Water treatment Plant Items}]{Water treatment Plant Items} \section[{Global ( a )}]{Global ( a )}\par 2011 \section[{August}]{August}\par The DP algorithm for functional and minimal cost design of water treatment system was developed in this research. The major inputs required are the design data on flow; raw-water suspended-solids concentration; alum dose; water temperature; and effective size, uniformity coefficient, and depth of filter sand. Data on unit costs of excavation, plain and reinforced cement VI. \section[{OPTIMIZATION ALGORITHM}]{OPTIMIZATION ALGORITHM}\par The sequential structure of serial systems can be exploited to transform the N-decision, one-state problems. This is accomplished by the procedure called dynamic programming, due to Richard \hyperref[b14]{Bellman (1957)}. Now by using there Return functions we can derive the Recursive equations for all of the unit processes. The general forms of recursive equations for unit processes are given in table 7.\par Table {\ref 7} : Recursive equations of different unit processes . -rapid mix, flocculation, sedimentation, and rapid send filters). Mathematical models describing the performance relationships and cost functions for component units of a conventional water-treatment system have been formulated. By combining these models, an algorithm using Dynamic programming and Taguchi Design of Experiments methods software was developed for functional and minimal design of the system. The proposed approach is tested with a case study. The conclusions are listed as follows: \section[{Recursive equations of Units Units}]{Recursive equations of Units Units}Q , A ] R .[ opt ) SS ( f f 1 2 * 1 = Filter s c 2 * 1 2 3 * 2 G , Q , A )] SS ( f R .[ opt ) SS ( f + = Sedimentation Q , T , G )] SS ( f R .[ opt ) SS ( f\par ? The computer times requirements for this model is very less than other similar model doing the same task.\par ? The optimal design results in savings of roughly 9.5\% in capital and annualized cost compared to the conventional design.\par ? The sensitivity analysis results signifies the objective function is more sensitive to design flow rate (Q), alum dose (A) and gravity sand filter head loss (H), respectively. Suspension characteristics such as particle size and distribution are of significance in the overall process of water treatment. There are some shortcomings in major requirements for the successful incorporation of PSD in the process design and selection. \hyperref[b0]{(Dharmappa, 1994)}. Future research dynamic programming \& Taguchi methods optimization of water treatment plant design may be directed to test the feasibility of considering PSD and molecular particle size distribution in water and further research in the field of software development with more levels for control factors.\begin{figure}[htbp] \noindent\textbf{2}\includegraphics[]{image-2.png} \caption{\label{fig_0}Fig. 2 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{1} \par \begin{longtable}{P{0.595\textwidth}P{0.255\textwidth}} State parameter\tabcellsep \\ Rapid Mix\tabcellsep Flocculation\\ SS 5\tabcellsep \end{longtable} \par \caption{\label{tab_0}Table 1 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{2} \par \begin{longtable}{} \end{longtable} \par \caption{\label{tab_1}Table 2 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{3} \par \begin{longtable}{} \end{longtable} \par \caption{\label{tab_3}Table 3 .}\end{figure} \begin{figure}[htbp] \noindent\textbf{3} \par \begin{longtable}{} \end{longtable} \par \caption{\label{tab_4}Table 3 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{P{0.051515151515151514\textwidth}P{0.1689217758985201\textwidth}P{0.448062015503876\textwidth}P{0.0005990133897110641\textwidth}P{0.002995066948555321\textwidth}P{0.021564482029598306\textwidth}P{0.017970401691331923\textwidth}P{0.002995066948555321\textwidth}P{0.0011980267794221282\textwidth}P{0.0017970401691331924\textwidth}P{0.0005990133897110641\textwidth}P{0.006589147286821705\textwidth}P{0.0005990133897110641\textwidth}P{0.003594080338266385\textwidth}P{0.03953488372093023\textwidth}P{0.011381254404510219\textwidth}P{0.002995066948555321\textwidth}P{0.00718816067653277\textwidth}P{0.0005990133897110641\textwidth}P{0.0005990133897110641\textwidth}P{0.002995066948555321\textwidth}P{0.012579281183932346\textwidth}P{0.0005990133897110641\textwidth}P{0.011980267794221284\textwidth}P{0.009584214235377026\textwidth}P{0.0023960535588442564\textwidth}P{0.0041930937279774485\textwidth}P{0.002995066948555321\textwidth}P{0.0011980267794221282\textwidth}P{0.0011980267794221282\textwidth}P{0.0023960535588442564\textwidth}P{0.0005990133897110641\textwidth}P{0.0017970401691331924\textwidth}P{0.0005990133897110641\textwidth}P{0.0017970401691331924\textwidth}P{0.0017970401691331924\textwidth}} \tabcellsep \multicolumn{22}{l}{Dynamic Programming and Taguchi Method Optimization of Water-Treatment-Plant Design}\\ \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \multicolumn{17}{l}{Annual Construction Cost(ACC)}\\ \tabcellsep Unit processes\tabcellsep \tabcellsep \tabcellsep \tabcellsep \multicolumn{18}{l}{Operation and Maintenance Cost}\tabcellsep ( O \& M)\\ 2011\tabcellsep Rapid mix\tabcellsep \multicolumn{21}{l}{Gr 1 ( . a 00251 ) f ) ( 181 ) 0007 DHR . 1 ( 988 . ) ( . 0 717 ) 0 . 0 CCI ( PR ) ( 79 . 0 799 . ) 0 Tr Tr × ) × Q ( Q 08507 ( 56947 . 0 . 1 = = ACC OM}\tabcellsep Gr\\ August\tabcellsep Alum dry stock\tabcellsep \multicolumn{5}{l}{10 A × ( 30366 . 3 0197 . 0 × = = ACC OM}\tabcellsep \multicolumn{2}{l}{4 Q ?}\tabcellsep \multicolumn{2}{l}{( )}\tabcellsep \multicolumn{4}{l}{656 A . 0 ×}\tabcellsep \multicolumn{2}{l}{849 ) CCI . 0 ) ( Q}\tabcellsep \multicolumn{5}{l}{994 ( . 0 PR}\tabcellsep )\tabcellsep . 0\tabcellsep 1847\tabcellsep (\tabcellsep PPI\tabcellsep )\tabcellsep 0\tabcellsep .\tabcellsep 0259\tabcellsep (\tabcellsep DHR\tabcellsep )\tabcellsep . 0\tabcellsep 743\\ 28\tabcellsep \tabcellsep ACC\tabcellsep =\tabcellsep . 642\tabcellsep Q ( 48\tabcellsep \multicolumn{2}{l}{×}\tabcellsep \multicolumn{2}{l}{Ts}\tabcellsep \multicolumn{2}{l}{)}\tabcellsep 0\tabcellsep \multicolumn{3}{l}{6916 .}\tabcellsep \multicolumn{2}{l}{CCI (}\tabcellsep )\tabcellsep 0\tabcellsep \multicolumn{3}{l}{992 .}\tabcellsep 1 (\tabcellsep ) 00383 .\tabcellsep Gs\tabcellsep (\tabcellsep a\tabcellsep f\tabcellsep )\\ Volume XI Issue V Version I\tabcellsep 1 B Flocculation 2 B Clarifier 0 B Filter structure Filter Media(single)\tabcellsep \multicolumn{21}{l}{PPI 756 ( 357 . 0 . 0 00047 . 1 ( ) ( ) n n U DHR PR ) 993 . 0 ( 181 785 . 0 . 0 ( ) ( ) a ) 1 0 006 399 . . ) ) f U CCI ) ( Ts 701 × 969 Q ( 040581 . . 0 . 0 ( ) ( 487 . 53 0 = ) ( 527 . 1872 c c PPI A A = ACC OM OM = 183 . 0 147 . 0 549 . 0 989 . 0 671 . 0 ) ( 1 61 0081 ) . . 0 ( ) ( ) ( ) ( 377 . 913 ) ( ) ( ) ( 11 . 13317 DHR PPI PR A OM a CCI A ACC f f f = = 0 = OM ) ( ) ( ) ( 3413 . 71 996 . 0 9336 . 0 = a CCI A ACC f f}\tabcellsep Gs\\ ( a ) Researches in Engineering\tabcellsep 3 B Filter backwash\tabcellsep \multicolumn{12}{l}{59 65 . 0 . ( ( 801 ) 0 7146 A f ) A 1 . 6 1 . 6 ( f . 0 ) . 0 ( ) ( 4533 127 . 108 ( 398 . 747 = = ACC OM ( 732 . 204 . 568}\tabcellsep \multicolumn{7}{l}{966 ( . 0 543 ) ACC . 0 ) PPI ( a PR ( 526 . 0 ) ( ) 982 . 0 ) ( )}\tabcellsep \multicolumn{2}{l}{) 219 . 0 315 . f ) 0}\tabcellsep (\tabcellsep DHR\tabcellsep )\tabcellsep . 0\tabcellsep 137\\ Journal of\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\ Global\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\ \tabcellsep Design variables\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\ \tabcellsep Where: R= Stage Returns\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\ \tabcellsep © 2011 Global Journals Inc. (US)\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par \caption{\label{tab_5}}\end{figure} \begin{figure}[htbp] \noindent\textbf{5} \par \begin{longtable}{P{0.5994103773584906\textwidth}P{0.07216981132075471\textwidth}P{0.1784198113207547\textwidth}} Item\tabcellsep \tabcellsep Value\\ Power(PR)\tabcellsep \tabcellsep 0.04\$/kWh\\ Labor Wage Rate(DHR)\tabcellsep \tabcellsep 11.00\$/hr\\ \multicolumn{2}{l}{Construction Cost Index/100(CCI)}\tabcellsep 3.25\\ Producer Price Index/100(PPI)\tabcellsep \tabcellsep 2.44\\ Alum\tabcellsep \tabcellsep 140\$/tone\\ Polymer\tabcellsep \tabcellsep 400\$/ tone\\ Polymer demand\tabcellsep \tabcellsep 6*10 -8 moles/m 2 of particle\\ Polymer molecular weight\tabcellsep \tabcellsep 5*10 4\\ each unit process be prepared with economic data. The\tabcellsep \multicolumn{2}{l}{assumed chemical costs are listed in}\\ values of parameters for determining Return functions\tabcellsep \\ that are used in the optimizations, as well as the\tabcellsep \end{longtable} \par \caption{\label{tab_6}Table 5 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{5} \par \begin{longtable}{} \end{longtable} \par \caption{\label{tab_7}Table 5 .}\end{figure} \begin{figure}[htbp] \noindent\textbf{6} \par \begin{longtable}{P{0.85\textwidth}} 2011\\ August\\ 29\\ Volume XI Issue v v Version I V\\ (a )\\ Journal of Researches in Engineering\\ Global\end{longtable} \par \caption{\label{tab_8}Table 6 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{4} \par \begin{longtable}{P{0.08630769230769231\textwidth}P{0.06276923076923077\textwidth}P{0.0026153846153846153\textwidth}P{0.007846153846153846\textwidth}P{0.015692307692307693\textwidth}P{0.010461538461538461\textwidth}P{0.0026153846153846153\textwidth}P{0.0026153846153846153\textwidth}P{0.010461538461538461\textwidth}P{0.005230769230769231\textwidth}P{0.007846153846153846\textwidth}P{0.007846153846153846\textwidth}P{0.0026153846153846153\textwidth}P{0.005230769230769231\textwidth}P{0.007846153846153846\textwidth}P{0.0026153846153846153\textwidth}P{0.0026153846153846153\textwidth}P{0.013076923076923078\textwidth}P{0.010461538461538461\textwidth}P{0.007846153846153846\textwidth}P{0.010461538461538461\textwidth}P{0.09676923076923077\textwidth}P{0.013076923076923078\textwidth}P{0.0026153846153846153\textwidth}P{0.007846153846153846\textwidth}P{0.0026153846153846153\textwidth}P{0.015692307692307693\textwidth}P{0.010461538461538461\textwidth}P{0.0026153846153846153\textwidth}P{0.007846153846153846\textwidth}P{0.020923076923076923\textwidth}P{0.013076923076923078\textwidth}P{0.0026153846153846153\textwidth}P{0.0026153846153846153\textwidth}P{0.02353846153846154\textwidth}P{0.06538461538461539\textwidth}P{0.026153846153846156\textwidth}P{0.02353846153846154\textwidth}P{0.018307692307692306\textwidth}P{0.03661538461538461\textwidth}P{0.026153846153846156\textwidth}P{0.041846153846153845\textwidth}P{0.02876923076923077\textwidth}P{0.007846153846153846\textwidth}P{0.015692307692307693\textwidth}P{0.015692307692307693\textwidth}P{0.013076923076923078\textwidth}P{0.0026153846153846153\textwidth}P{0.0026153846153846153\textwidth}P{0.0026153846153846153\textwidth}P{0.015692307692307693\textwidth}} 1 R\tabcellsep \multicolumn{3}{l}{=}\tabcellsep \multicolumn{4}{l}{. 8636}\tabcellsep \multicolumn{6}{l}{( 59}\tabcellsep \multicolumn{5}{l}{f A}\tabcellsep \multicolumn{2}{l}{. )}\tabcellsep \multicolumn{2}{l}{671}\tabcellsep \multicolumn{3}{l}{+}\tabcellsep \multicolumn{5}{l}{. 4688}\tabcellsep \multicolumn{3}{l}{( 94}\tabcellsep f A\tabcellsep . )\tabcellsep 546\tabcellsep +\tabcellsep . 46\tabcellsep ( 877\tabcellsep f A\tabcellsep . )\tabcellsep 9336\tabcellsep +\\ \multicolumn{3}{l}{. 293}\tabcellsep \multicolumn{3}{l}{847}\tabcellsep (\tabcellsep \multicolumn{2}{l}{A}\tabcellsep f\tabcellsep \multicolumn{2}{l}{)}\tabcellsep .\tabcellsep \multicolumn{5}{l}{59}\tabcellsep \multicolumn{2}{l}{+}\tabcellsep \multicolumn{5}{l}{2006}\tabcellsep \multicolumn{2}{l}{.}\tabcellsep \multicolumn{3}{l}{( 13}\tabcellsep A\tabcellsep f\tabcellsep )\tabcellsep .\tabcellsep 65\tabcellsep +\tabcellsep . 363\tabcellsep ( 86\tabcellsep A\tabcellsep f\tabcellsep )\tabcellsep .\tabcellsep 801\tabcellsep +\tabcellsep 119\tabcellsep .\tabcellsep ( 195\tabcellsep A\tabcellsep f\tabcellsep )\tabcellsep 7146 .\\ \multicolumn{2}{l}{2 R}\tabcellsep \multicolumn{3}{l}{=}\tabcellsep \multicolumn{4}{l}{1223}\tabcellsep .\tabcellsep \multicolumn{5}{l}{52}\tabcellsep \multicolumn{3}{l}{(}\tabcellsep \multicolumn{2}{l}{c A}\tabcellsep \tabcellsep )\tabcellsep \multicolumn{2}{l}{701 .}\tabcellsep \multicolumn{4}{l}{( U}\tabcellsep n\tabcellsep 1 )\tabcellsep \multicolumn{4}{l}{00047 .}\tabcellsep +\tabcellsep 599\tabcellsep .\tabcellsep 165\tabcellsep (\tabcellsep c A\tabcellsep )\tabcellsep 469 .\tabcellsep ( U\tabcellsep n\tabcellsep 1 )\tabcellsep 006 .\\ Alum feed \& Rapid mix\tabcellsep \multicolumn{10}{l}{54067 . . ( 12662 = 4 .}\tabcellsep \multicolumn{5}{l}{( )}\tabcellsep .\tabcellsep \multicolumn{3}{l}{656 .}\tabcellsep \tabcellsep \multicolumn{13}{l}{r 00160934 ) 10007 ( . 79 . 0 + )}\tabcellsep + (\tabcellsep 849 3817 . . 0 ) .\tabcellsep (\tabcellsep .\tabcellsep )\tabcellsep 794 .\tabcellsep . 1 (\tabcellsep ) 00251\tabcellsep +\end{longtable} \par \caption{\label{tab_9}Table 4 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{8} \par \begin{longtable}{P{0.29475806451612907\textwidth}P{0.23306451612903223\textwidth}P{0.3221774193548387\textwidth}} \tabcellsep EPA WaterTreatment Plant\tabcellsep 1 B Suggested Model Water Treatment Plant\\ Total Annualized Cost(\$)\tabcellsep 1046570\tabcellsep 955757\\ Reduction(\%)\tabcellsep 9.5\tabcellsep \\ VIII.\tabcellsep \tabcellsep \end{longtable} \par \caption{\label{tab_10}Table 8 :}\end{figure} \footnote{© 2011 Global Journals Inc. (US)} \backmatter \subsection[{CONCLUSION}]{CONCLUSION}\par So far very few works have been done on the optimization of a water treatment system. The scope of this research was restricted to the economic optimization of conventional water treatment systems for\par The results signifies the objective function is more sensitive to Design flow rate (Q), alum dose (A) and gravity sand filter head loss (H) respectively. If the design flow is varied in the range of ±20\%, the total annualized cost will vary +12.6\% and 15.2 \% A practical approach to analyzing either uncertain or variable situation is to test the sensitive of the optimal design to variation in key system parameters. \begin{bibitemlist}{1} \bibitem[ J. Envir. Engrg. Div]{b12}\label{b12} \textit{}, \textit{J. Envir. Engrg. Div} ASCE. 108 (5) p. . \bibitem[Hillier and Research ()]{b5}\label{b5} \textit{}, F S Hillier , Operations Research . 1974. Holden-Day, Inc. Second Edition. \bibitem[Dharmappa et al. ()]{b0}\label{b0} ‘a), "water treatment system design for turbidity removal’. H B Dharmappa , O Fujiwara , J Verink , S Vingneswaran . \textit{II; optimization, j. Envir. Engrg., ASCE} 1994. 120 (4) p. . \bibitem[Hinomoto ()]{b10}\label{b10} ‘Capital cost functions of a surface water treatment plant and its components’. H Hinomoto . \textit{water resour. Res} 1977. 13 (6) p. . \bibitem[Clark ()]{b11}\label{b11} \textit{Cost estimating for conventional water treatment}, R M Clark . 1982. \bibitem[Walski ()]{b15}\label{b15} \textit{Discussion of preliminary optimal design model for wastewater quality and treatment}, T M Walski . 1991. 1990. New York , N.Y.: Mc Graw-Hill Book Co. , Inc. p. 177. (A handbook of community water supplies. 4th Ed., F. W. Pontius, ed.) \bibitem[Bellman ()]{b14}\label{b14} \textit{Dynamic programming}, R Bellman . 1957. Princeton, N.J: Princeton university press. \bibitem[Mhaisalkar et al. ()]{b1}\label{b1} ‘Dynamic programming optimization of water treatment plant design’. V A Mhaisalkar , K Bassin , P Paramasivam , Khanna . \textit{J. Envir. Engrg} 1993. 119 (6) p. . \bibitem[Gumerman et al. ()]{b13}\label{b13} \textit{Estimating water treatment costs}, R C Gumerman , R L Clup , S P Hansen . 1979. Cincinnati, Ohio. 2. Municipal Environmental Research Laboratory, U.S. Environmental Protection Agency (cost curves applicable to 1 to 200 mgd treatment plants) \bibitem[Eldred et al.]{b17}\label{b17} ‘Formulations for surrogate-based optimization under uncertainty’. M S Eldred , A A Giunta , S F Wojtkiewicz , Jr , T G Trucano , \textunderscore 2002\textunderscore . \textit{Proc., 9th AIAA/ISSMO Symp. on Multidisciplinary Analysis and Optimization}, (9th AIAA/ISSMO Symp. on Multidisciplinary Analysis and OptimizationAtlanta) (Paper AIAA-2002-5585) \bibitem[Desmond et al. ()]{b18}\label{b18} ‘Integral Water Treatment Plant Modeling: Improvements for Particle Processes’. F Desmond , Jeffrey A Lawler , Nason . \textit{Environ. Sci. Technol} 2005. (17) p. . \bibitem[Taguchi ()]{b9}\label{b9} \textit{Introduction to Quality Engineering}, G Taguchi . 1986. Dearborn, Mi: American Supplier Institute Inc. \bibitem[Kackar (1985)]{b7}\label{b7} ‘Off-Line Quality Control, Parameter Design, and the Taguchi method’. Raghu Kackar . \textit{journal of Quality Technology} August 9. 1985. 17 (4) p. . \bibitem[Wiesner ()]{b2}\label{b2} \textit{Optimal water treatment plant configuration: effect of raw water characteristics}, M R Wiesner . 1985. Baltimore, Md. Johns Hopkins University (Ph.D. thesis) \bibitem[Wiesner ()]{b3}\label{b3} ‘Optimal water treatment plant design’. M R Wiesner . \textit{Journal of Environmental Engineering division} 1987. ASCE. 113 (3) p. . \bibitem[Letterman and Iyer ()]{b4}\label{b4} ‘Process model application in potable water treatment’. R D Letterman , D R Iyer . \xref{http://dx.doi.org/10.1021/es050089e\#es050089eAF1\#es050089eAF1}{10.1021/es050089e\#es050089eAF1\#es050089eAF1}. \textit{Proc. En grg. Foundation Conf. On theory, Practice, and process principles}, (En grg. Foundation Conf. On theory, Practice, and process principles) 1977. p. . (For physical separation, pacific Grove, calif) \bibitem[Phadke ()]{b8}\label{b8} \textit{Quality Engineering Using Robust Design}, S M Phadke . 1989. Englewood Cliffs, N.j: Prentice Hall. \bibitem[Li et al.]{b16}\label{b16} ‘Settling and coagulating behavior of fractal aggregates’. X.-Y Li , B E Logan , \textunderscore 2000\textunderscore . \textit{Water Sci. Technol} 42 (3) p. . \bibitem[Asce and Csse ()]{b6}\label{b6} \textit{water treatment plant design}, Awwa Asce , Csse . 1990. New York: American water work association, Inc. \end{bibitemlist} \end{document}