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\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{2011-02-15 (revised: 15 February 2011)} \def\TheID{\makeatother } \def\TheDate{2011-02-15} \title{Fuzzy Goal Programming Method for SolvingMulti-Objective Transportation Problems} \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]{S G Acharyulu} \author[2]{Dr. K V V Chandra Mouli} \affil[1]{ GITAM Institute of Technology, GITAM University} \renewcommand\Authands{ and } \date{\small \em Received: 13 January 2011 Accepted: 3 February 2011 Published: 15 February 2011} \maketitle \begin{abstract} The multi-objective transportation problem refers to a special class of vector minimum linear programming problem, in which constraints are of inequality type and all the objectives are non-commensurable and conflict with each other. A common problem encountered in solving such multi-objective problems is that to identify a compromise solution among a large number of non-dominated solutions, the decision maker has to develop a utility function for meeting the desired goal. In this paper, fuzzy membership functions are considered and deviation goals also taken for each objective function. Fuzzy max-min operator is implemented to show the effectiveness of the proposed methodology. LINGO software package is used to solve constrained optimization problem. To illustrate the proposed method, two numerical examples are solved and the results have been compared with interactive, fuzzy and deviation criterion approaches. \end{abstract} \keywords{} \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 classical transportation problem is one of the sub classes of linear programming problem in which all constraints are inequality type. \hyperref[b7]{Hitchcock (1941)} developed transportation model. Because of the complexity of the social and economic environment requires explicit consideration of criteria other than cost, the single objective transportation problems in real world cases can be formulated as multi-objective models. \hyperref[b3]{Charnes and Cooper (1961)} first discussed on various approaches to solutions of managerial level problems involving multiple conflicting objectives. \hyperref[b9]{Ignizio (1978)} applied goal programming for multiobjective optimization problems and solved twoobjective optimization problem. Some of the authors (see \hyperref[b5]{Garfinkl \& Rao 1971}; \hyperref[b20]{Swaroop et al., 1976}) have solved the two objective problem by giving high and low priorities to the objectives. \hyperref[b0]{Belenson and Kapur (1973)} presented two person-zero sum game approach consists of a p x p pay off matrix and solved each objective function individually finally developed best compromise solution using proper weights to the objective functions. Jimmenez and Vudegay (1999) solved a multi-criteria transportation problem using parametric approach by developing auxiliary solutions. Rakesh \hyperref[b16]{Varma et al., (1997)} used fuzzy min operator approach to develop a compromise solution for the multi-objective problem. \hyperref[b18]{Ringuest and Rinks (1987)} proposed two interactive algorithms for generating all non-dominated solutions and identified minimum cost solution as a best compromise solution. Gen et al., (1998) solved a bi-criteria transportation problem using hybrid genetic algorithm adopting spanning tree based prufer number to generate all possible basic solutions. Waiel. ( {\ref 2001}) developed all non-dominated solutions and defined family of distance function to arrive a compromise solution.\par The existing procedures in the literature (see \hyperref[b4]{Deb, 2003;}\hyperref[b17]{Rao, 2003)} for solving multi-objective transportation problems can be divided into two categories. First category of those are generating all the sets of efficient solutions (see \hyperref[b18]{Ringuest and Rinks, 1987;}\hyperref[b6]{Gen et al., 1997)} and the second category represents the procedure of using an additional criterion to obtain the best compromise solution among the set of efficient solutions (see Rakesh \hyperref[b16]{Varma \& Biswas, 1997;} {\ref Gen et al., 1998;}\hyperref[b2]{Bit et al., 1992}; and Sy-Ming Gun \& Yan -Kuen Wu, 1999) developed various functions to achieve direct compromise solution without developing and testing all the Pareto solutions.\par Although several researchers have been proposed various advances in transportation problems ( see \hyperref[b1]{Bit et al.,1993} In this paper, authors propose membership functions and goal deviation functions from Pareto solutions for each objective, and these functions are added as constraints. By introducing a max-min operator ? an auxiliary variable, then the equivalent fuzzy interactive goal programming problem is formulated to maximize ? and the solution is obtained by using LINGO software. The remaining of the paper is organized as follows: in section 2 we give a mathematical model of the multi-objective transportation problem (MOTP) and formulation with fuzzy max-min operator and goal deviations. Section 3 ( ) [ ] ( ) L U X F U X F k k k k k ? ? ? ? ? = µ\par represents proposed methodology; while in section 4 two numerical examples are solved. Finally, in section 5 and 6 we discuss on the results and conclusions. \section[{II.}]{II.} \section[{MATHEMATICAL MODEL}]{MATHEMATICAL MODEL}\par In a typical transportation problem, a homogenous product is to be transported from several origins (or sources) to numerous destinations in such way that the total transportation cost is minimum. Suppose there are "m" origins (i=1,2,??,m) and "n" destinations (j=1,2,??.,n). The sources may be production facilities, warehouses etc and they are characterized by available supplies a 1 , a 2, ?,a m . The destinations may be warehouses and sales outlets etc, and they are characterized by demand levels b 1 , b 2 ,?.,b n . A penalty c ij is associated with transporting a unit of product from origin i to destination j. The penalty could represent transportation cost, delivery time, distance, quality of goods delivered under used capacity or many other criteria. A variable x ij is used to represent the unknown quantity to be transported from origin O i to destination D j . In the real life, however all transportation problems are not single objective. The transportation problems, which are characterized by multiple objective functions, are considered in this paper. The decision maker would like to minimize the set of K objectives simultaneously; a point will likely be reached where a further reduction of the value of any single objective function may only be obtained at the expense of increasing the value of at least one other objective function. Thus, in general, the objectives will also be conflicting. The mathematical model of the multi-objective transportation problem is written as follows:\par (2.1) X C (X) F Min ij ij k n 1 j m 1 i k ? ? = = = subject to the (2.4) j and i all for , 0 x (2.3) n, ........, 1,2,...... j , b x (2.2) m, ........, 1,2,...... i , a x ij j m 1 i ij i n 1 j ij ? = = = ? ? ? = = Where, \{ \} (X) F ....., .......... (X), F (X), F (X) F k 2 1 k \section[{=}]{=}\par is a vector of K objective functions and superscript on both F k (X) and C k ij are used to identify the number of objective functions (k=1,2,?.,K) without loss of generality it will be assumed in the whole paper that a i ?0 and b j ?0 for all i and j and ? i a i =? j b j , c ij >0 for all i and j. \section[{a) Problem Formulation Using Fuzzy Max-Min Operator}]{a) Problem Formulation Using Fuzzy Max-Min Operator}\par Fuzzy set theory appears to be an ideal approach to deal with decision problems that are formulated as linear programming models with imprecision parameters. Two face fuzzy linear programming models are designated by Sy-Ming Gun \& Yan-Kuen Wu (1999) for such problems. In the literature fuzzy linear programming has been classified into different categories, depending on how imprecise parameters are modeled by possibility distributions or subjective preference based membership functions. In this paper the net relative deviation is considered as fuzzy variable and converted into deterministic form using Zadeh's max-min operator as per \hyperref[b23]{Zimmermann (1985)}. We define a linear membership function by considering suitable upper and lower bounds to the objective function as given below.µ[F k (X)] = 1, if F k (X) ? U k ,if Lk