2011
8
4
4
117
Cover Special Issue vol. 8, no. 4, October 2011
2
2
1

0
0
SOME PROPERTIES FOR FUZZY CHANCE CONSTRAINED
PROGRAMMING
SOME PROPERTIES FOR FUZZY CHANCE CONSTRAINED
PROGRAMMING
2
2
Convexity theory and duality theory are important issues in math
ematical programming. Within the framework of credibility theory, this paper
rst introduces the concept of convex fuzzy variables and some basic criteria.
Furthermore, a convexity theorem for fuzzy chance constrained programming
is proved by adding some convexity conditions on the objective and constraint
functions. Finally, a duality theorem for fuzzy linear chance constrained pro
gramming is proved.
1
Convexity theory and duality theory are important issues in math
ematical programming. Within the framework of credibility theory, this paper
rst introduces the concept of convex fuzzy variables and some basic criteria.
Furthermore, a convexity theorem for fuzzy chance constrained programming
is proved by adding some convexity conditions on the objective and constraint
functions. Finally, a duality theorem for fuzzy linear chance constrained pro
gramming is proved.
1
8
Xiaohu
Yang
Xiaohu
Yang
Department of Statistics, Xi'an University of Finance and Economics,
Xi'an 710061, China
Department of Statistics, Xi'an University
China
yxh12@163.com
Convexity theorem
Duality theorem
Fuzzy variable
Chance con strained programming
[bibitem{Bellman and Zadeh}R. E. Bellman and L. A. Zadeh, {it Decision making in a fuzzy environment},##Management Science, {bf textbf{17}} (1970), 141164. ##bibitem{Delgado}M. Delgado, J. Kacprzyk, J. L. Verdegay and M. A. Vila, {it Fuzzy optimization: recent advances},##PhysicaVerlag, Wurzburg, 1994. ##bibitem{Ji}X. Ji X and Z. Shao, {it Model and algorithm for bilevel newsboy##problem with fuzzy demands and discounts}, Applied Mathematics and##Computation, {bf textbf{172(1)}} (2006), 163174. ##bibitem{Lai 1}Y. J. Lai and C. L. Hwang, {it Fuzzy mathematical programming: methods and applications},##Lecture Notes in Economics and Mathematical Systems,## Springer, Berlin, {bftextbf{394}} (1992). ##bibitem{Lai 2}Y. J. Lai and C. L. Hwang, {it Fuzzy multiple objective decision making: methods and applications},##Lecture notes in Economics and Mathematical Systems,## Springer, Berlin, {bftextbf{404}} (1992). ##bibitem{Li 2006}X. Li and B. Liu, {it A sufficient and necessary condition for credibility measures},##International Journal of Uncertainty, Fuzziness & KnowledgeBased##Systems, {bf textbf{14(5)}} (2006), 527535. ##bibitem{Li and liu}X. Li and B. Liu, {it The independence of fuzzy variables with##applications}, International Journal of Natural Sciences and##Technology, {bf textbf{1(1)}} (2006), 95100. ##bibitem{Li2010}X. Li, Z. Qin and L. X. Yang, {it A chanceconstrained portfolio selection model with risk##constraints}, Applied Mathematics and Computation, {bf##textbf{217}} (2010), 949951. ##bibitem{Li2011}X. Li, Z. Qin, L. X. Yang and K. P. Li, {it Entropy maximization model for trip distribution problem with##fuzzy and random parameters}, Journal of Computational and Applied##Mathematics, doi:10.1016/j.cam.2010.09.004. ##bibitem{Liu maximax ccp}B. Liu and K. Iwamura, {it Chance constrained programming with fuzzy parameters},##Fuzzy Sets and Systems, {bf textbf{94(2)}} (1998), 227237. ##bibitem{Liu minimax ccp}B. Liu, {it Minimax chance constrained programming models for fuzzy decision systems},##Information Sciences, {bf textbf{112(14)}} (1998), 2538. ##bibitem{Liu dcp}B. Liu, {it Dependentchance programming in fuzzy environment}, Fuzzy Sets and Systems, {bf textbf{109(1)}} (2000), 97106. ##bibitem{Liu 2002}B. Liu, {em Theory and practice of uncertain programming}, PhysicaVerlag, Heidelberg, 2002. ##bibitem{Liu and Liu Evofv}B. Liu and Y. K. Liu, {it Expected value of fuzzy variable and fuzzy expected value models},##IEEE Transactions on Fuzzy Systems, {bf textbf{10(4)}} (2002),##bibitem{Liu 2004}B. Liu, {em Uncertainty theory: an introduction to its axiomatic##foundations}, SpringerVerlag, Berlin, 2004. ##bibitem{Liu 2007}B. Liu, {em Uncertainty theory}, 2nd ed.,##SpringerVerlag, Berlin, 2007. ##bibitem{Liu and Gao independent}Y. K. Liu and J. Gao, {it The independence of fuzzy variables with applications to fuzzy random##optimization}, International Journal of Uncertainty, Fuzziness &##KnowledgeBased Systems, {bf textbf{15(Suppl.2)}} (2007), 120. ##bibitem{peng and liu}J. Peng and B. Liu, {it Parallel machine scheduling models with fuzzy processing times},##Information Sciences, {bf textbf{166(14)}} (2004), 4966. ##bibitem{slowinski}R. Slowinski, {em Fuzzy sets in decision analysis,##operations research and statistics}, Kluwer Academic Publishers,##Dordrecht, 1998. ##bibitem{vajda}S. Vajda, {em Probabilistic programming}, Academic Press,##New York, 1972. ##bibitem{Zadeh 1978}L. A. Zadeh, {it Fuzzy sets as a basis for a theory of possibility}, Fuzzy Sets and##Systems, {bf textbf{1}} (1978), 328. ##bibitem{Zadeh 1979}L. A. Zadeh, {it A theory of approximate reasoning}, In: Hayes J et al.,##Mathematical Frontiers of the Social and Policy Sciences, Westview##Press, Boulder, Cororado, (1979), 69129. ##bibitem{zhao and liu}R. Zhao and B. Liu, {it Standby redundancy optimization problems with fuzzy lifetimes},##Computers & Industrial Engineering, {bf textbf{149(2)}} (2005),##bibitem{Zheng and Liu}Y. Zheng and B. Liu, {it Fuzzy vehicle routing model with##credibility measure and its hybrid intelligent algorithm}, Applied##Mathematics and Computation, {bf textbf{176(2)}} (2006), 673683. ##bibitem{Zhou}J. Zhou and B. Liu B, {it Modeling capacitated locationallocation##problem with fuzzy demands}, Computers & Industrial Engineering,##{bf textbf{53(3)}} (2007), 454468.##]
ALGORITHMS FOR BIOBJECTIVE SHORTEST PATH
PROBLEMS IN FUZZY NETWORKS
ALGORITHMS FOR BIOBJECTIVE SHORTEST PATH
PROBLEMS IN FUZZY NETWORKS
2
2
We consider biobjective shortest path problems in networks with
fuzzy arc lengths. Considering the available studies for single objective shortest
path problems in fuzzy networks, using a distance function for comparison of
fuzzy numbers, we propose three approaches for solving the biobjective prob
lems. The rst and second approaches are extensions of the labeling method to
solve the single objective problem and the third approach is based on dynamic
programming. The labeling methods usually producing several nondominated
paths, we propose a fuzzy number ranking method to determine a fuzzy short
est path. Illustrative examples are worked out to show the eectiveness of our
algorithms.
1
We consider biobjective shortest path problems in networks with
fuzzy arc lengths. Considering the available studies for single objective shortest
path problems in fuzzy networks, using a distance function for comparison of
fuzzy numbers, we propose three approaches for solving the biobjective prob
lems. The rst and second approaches are extensions of the labeling method to
solve the single objective problem and the third approach is based on dynamic
programming. The labeling methods usually producing several nondominated
paths, we propose a fuzzy number ranking method to determine a fuzzy short
est path. Illustrative examples are worked out to show the eectiveness of our
algorithms.
9
37
Iraj
Mahdavi
Iraj
Mahdavi
Department of Industrial Engineering, Mazandaran University of Sci
ence & Technology, Babol, Iran
Department of Industrial Engineering, Mazandaran
Iran
irajarash@rediffmail.com
Nezam
MahdaviAmiri
Nezam
MahdaviAmiri
Faculty of Mathematical Sciences, Sharif University of Tech
nology, Tehran, Iran
Faculty of Mathematical Sciences, Sharif
Iran
nezamm@sharif.edu
Shahrbanoo
Nejati
Shahrbanoo
Nejati
Department of Industrial Engineering, Mazandaran University
of Science & Technology, Babol, Iran
Department of Industrial Engineering, Mazandaran
Iran
nejati sh@yahoo.com
Biobjective shortest path
Fuzzy network
Labeling method
Dynamic programming
Fuzzy ranking methods
[bibitem{baas1977rating}##S. M. Baas and H.~Kwakernaak, {it Rating and ranking of multipleaspect alternatives using fuzzy## sets}, Automatica, {bf 13}textbf{(1)} (1977), 4758. ##bibitem{baldwin1979comparison}##J. F. Baldwin and N. C. F Guild, {it Comparison of fuzzy sets on the same decision space}, Fuzzy Sets and Systems, {bf 2}textbf{(3)} (1979), 213231. ##bibitem{bellman1958routing}##R.~Bellman, {it On a routing problem, Quart}, J. Appl. Math, {bf 16}textbf{(1)} (1958), 8790. ##bibitem{bortolan1985review}##G.~Bortolan and R.~Degani, {it A review of some methods for ranking fuzzy subsets}, Fuzzy Sets and Systems, {bf 15}textbf{(1)} (1985), 119. ##bibitem{brumbaugh1989empirical}##D.~BrumbaughSmith and D. Shier, {it An empirical investigation of some bicriterion shortest path## algorithms}, European Journal of Operational Research, {bf 43}textbf{(2)} (1989), 216224. ##bibitem{campos1989subjective}##L.~Campos and A.~Munoz, {it A subjective approach for ranking fuzzy numbers}, Fuzzy Sets and Systems, {bf 29}textbf{(12)} (1989), 145153. ##bibitem{chanas1995fuzzy}##S.~Chanas, M.~Delgado, J. L. Verdegay, and M. A. Vila, {it Fuzzy optimal flow on imprecise structures}, European Journal of Operational Research, {bf 83}textbf{(3)} (1995), 568580. ##bibitem{chang1981ranking}##W.~Chang, {it Ranking of fuzzy utilities with triangular membership functions}, In Proc. Int. Conf. of Policy Anal. and Inf. Systems, (1981), 263272. ##bibitem{chen1985ranking}##S. H. Chen, {it Ranking fuzzy numbers with maximizing set and minimizing set}, Fuzzy Sets and Systems, {bf 17}textbf{(2)} (1985), 113129. ##bibitem{choobineh1993index}##F.~Choobineh and H.~Li, {it An index for ordering fuzzy numbers}, Fuzzy Sets and Systems, {bf 54}textbf{(3)} (1993), 287294. ##bibitem{delgado1990valuation}##M.~Delgado, J. L. Verdegay and M. A. Vila, {it On valuation and optimization problems in fuzzy graphs: a general## approach and some particular cases}, INFORMS Journal on Computing, {bf 2}textbf{(1)} (1990), 74. ##bibitem{dijkstra1959note}##E. W. Dijkstra, {it A note on two problems in connexion with graphs}, Numerische mathematik, {bf 1}textbf{(1)} (1959), 269271. ##bibitem{dreyfus1969appraisal}##S. E. Dreyfus, {it An appraisal of some shortestpath algorithms}, Operations Research, {bf 17}textbf{(3)} (1969), 395412. ##bibitem{dubois1978algorithmes}##D.~Dubois and H.~Prade, {it Algorithmes de plus courts chemins pour traiter des donnees floues.## RAIRORecherche Op{'e}rationnelle}, Operations Research, {bf 12}(1978), 212227. ##bibitem{dubois1980fuzzy}##D.~Dubois and H.~Prade, {it Fuzzy sets and systems: theory and applications}, Academic Pr, 1980. ##bibitem{furukawa1994parametric}##N.~Furukawa, {it A parametric total order on fuzzy numbers and a fuzzy shortest route## problem}, Optimization, {bf 30}textbf{(4)} (1994), 367377. ##bibitem{hansen1980bicriterion}##P.~Hansen, {it Bicriterion path problems}, In Multiple criteria decision making: theory and application:## proceedings of the third conference, Hagen/K{`e}onigswinter, West Germany,## August 2024, (1979), 109, Springer, 1980. ##bibitem{helgason1995primal}##R.V. Helgason and J. L. Kennington, {it Primal simplex algorithms for minimum cost network flows}, Handbooks in Operations Research and Management Science, {bf 7} (1995), 85133. ##bibitem{huarng1996computational}##F.~Huarng, P. Pulat, and L. S. Shih, {it A computational comparison of some bicriterion shortest path## algorithms}, Journal of the Chinese Institute of Industrial Engineers, {bf 13}textbf{(2)} (1996), 121125. ##bibitem{klein1991fuzzy}##C. M. Klein, {it Fuzzy shortest paths}, Fuzzy Sets and Systems, {bf 39}textbf{(1)} (1991), 2741. ##bibitem{k—czy1992fuzzy}##L. T. K{'o}czy, {it Fuzzy graphs in the evaluation and optimization of networks}, Fuzzy Sets and Systems, {bf 46}textbf{(3)} (1992), 307319. ##bibitem{lin1993fuzzy}##K. C. Lin and M. S. Chern, {it The fuzzy shortest path problem and its most vital arcs}, Fuzzy Sets and Systems, {bf 58}textbf{(3)} (1993), 343353. ##bibitem{liou1992ranking}##T. S. Liou and M. J. J. Wang, {it Ranking fuzzy numbers with integral value}, Fuzzy Sets and Systems, {bf 50}textbf{(3)} (1992), 247255. ##bibitem{mahdavi2009dynamic}##I.~Mahdavi, R.~Nourifar, A.~Heidarzade and N. M. Amiri, {it A dynamic programming approach for finding shortest chains in a fuzzy network}, Applied Soft Computing, {bf 9}textbf{(2)} (2009), 503511. ##bibitem{martins1984multicriteria}##E. Q. V. Martins, {it On a multicriteria shortest path problem}, European Journal of Operational Research, {bf 16}textbf{(2)} (1984), 236245. ##bibitem{ishwar1991parametric}##J.~Mote, I.Murthy and D. L. Olson {it A parametric approach to solving bicriterion shortest path problems}, European Journal of Operational Research, {bf 53}textbf{(1)} (1991), 8192. ##bibitem{namorado1982bicriterion}##J. C. Namorado~Climaco and E.~Queiros Vieira~Martins, {it A bicriterion shortest path algorithm}, European Journal of Operational Research, {bf 11}textbf{(4)} (1982), 399404. ##bibitem{okada2000shortest}##S.~Okada and T.~Soper, {it A shortest path problem on a network with fuzzy arc lengths}, Fuzzy Sets and Systems, {bf 109}textbf{(1)} (2000), 129140. ##bibitem{prade1979using}##H.~Prade, {it Using fuzzy set theory in a scheduling problem: a case study}, Fuzzy Sets and Systems, {bf 2}textbf{(2)} (1979), 153165. ##bibitem{przybylski2008two}##A.~Przybylski, X.~Gandibleux and M.~Ehrgott, {it Two phase algorithms for the biobjective assignment problem}, European Journal of Operational Research, {bf 185}textbf{(2)} (2008), 509533. ##bibitem{ram’k1985inequality}##J. Ram{'i}k and J. Rimanek, {it Inequality relation between fuzzy numbers and its use in fuzzy optimization}, Fuzzy Sets and Systems, {bf 16}textbf{(2)} (1985), 123138. ##bibitem{saade1992ordering}##J. J. Saade and H.~Schwarzlander, {it Ordering fuzzy sets over the real line: an approach based on## decision making under uncertainty}, Fuzzy Sets and Systems, {bf 50}textbf{(3)} (1992), 237246. ##bibitem{sadeghpour2001distance}##B.~Sadeghpour~Gildeh and D.~Gien, {it La distanceDp, q et le cofficient de corr{'e}lation entre deux## variables al{'e}atoires floues}, Actes de LFA, (2001), 97102. ##bibitem{serafini1987some}##P.~Serafini, {it Some considerations about computational complexity for multi## objective combinatorial problems}, In Recent Advances and Historical Development of Vector## Optimization: Proceedings of an International Conference of Vector## Optimization Held at the Technical University of Darmstadt, FRG, August 47,## (1986), 222, Springer, 1987. ##bibitem{skriver2000label}##A. J. V. Skriver and K. A. Andersen, {it A label correcting approach for solving bicriterion shortestpath problems}, Computers & Operations Research, {bf 27}textbf{(6)} (2000), 507524. ##bibitem{skriver2000classification}##A. J. V. Skriver, {it A classification of bicriterion shortest path (BSP) algorithms}, Asia Pacific Journal of Operational Research, {bf 17}textbf{(2)} (2000), 199212. ##bibitem{TajdinMahdavi2010}##A.~Tajdin, I.~Mahdavi, N.~MahdaviAmiri and B.~SadeghpourGildeh, {it Computing a fuzzy shortest path in a network with mixed fuzzy arc lengths using $alpha$ cuts}, Computer and Mathematics with Applications, 2010. ##bibitem{tung1988bicriterion}##C. T. Tung and K. L. Chew, {it A bicriterion Paretooptimal path algorithm}, ASIAPACIFIC J. OPER. RES., {bf 5}textbf{(2)} (1988), 166172. ##bibitem{tung1992multicriteria}##C.~Tung and K.~Lin~Chew, {it A multicriteria Paretooptimal path algorithm}, European Journal of Operational Research, {bf 62}textbf{(2)} (1992), 203209. ##bibitem{wang1997comparative}##X.~Wang, {it A comparative study of the ranking methods for fuzzy quantities}, Ghent University, Ghent, 1997. ##bibitem{wang2001reasonable}##X.~Wang and E. E. Kerre, {it Reasonable properties for the ordering of fuzzy quantities (I)}, Fuzzy Sets and Systems, {bf 118}textbf{(3)} (2001), 375385. ##bibitem{wang2001reasonable2}##X.~Wang and E. E. Kerre, {it Reasonable properties for the ordering of fuzzy quantities (II)}, Fuzzy Sets and Systems, {bf 118}textbf{(3)} (2001), 387405. ##bibitem{yager1980general}##R. A. Yager, {it On a general class of fuzzy connectives}, Fuzzy sets and Systems, {bf 4}textbf{(3)} (1980), 235242. ##bibitem{yager1986paths}##R. A. Yager, {it Paths of least resistance in possibilistic production systems}, Fuzzy Sets and Systems, {bf 19}textbf{(2)} (1986), 121132. ##bibitem{zadeh1965fuzzy}##L. A. Zadeh, {it Fuzzy sets}, Information and control, {bf 8}textbf{(3)} (1965), 338353.##]
A FUZZY MINIMUM RISK MODEL FOR THE RAILWAY
TRANSPORTATION PLANNING PROBLEM
A FUZZY MINIMUM RISK MODEL FOR THE RAILWAY
TRANSPORTATION PLANNING PROBLEM
2
2
The railway transportation planning under the fuzzy environment
is investigated in this paper. As a main result, a new modeling method, called
minimum risk chanceconstrained model, is presented based on the credibility
measure. For the convenience ofs olving the mathematical model, the crisp
equivalents ofc hance functions are analyzed under the condition that the
involved fuzzy parameters are trapezoidal fuzzy variables. An approximate
model is also constructed for the problem based on an improved discretization
method for fuzzy variables and the relevant convergence theorems. To
obtain an approximate solution, a tabu search algorithm is designed for the
presented model. Finally, some numerical experiments are performed to show
the applications ofthe model and the algorithm.
1
The railway transportation planning under the fuzzy environment
is investigated in this paper. As a main result, a new modeling method, called
minimum risk chanceconstrained model, is presented based on the credibility
measure. For the convenience ofs olving the mathematical model, the crisp
equivalents ofc hance functions are analyzed under the condition that the
involved fuzzy parameters are trapezoidal fuzzy variables. An approximate
model is also constructed for the problem based on an improved discretization
method for fuzzy variables and the relevant convergence theorems. To
obtain an approximate solution, a tabu search algorithm is designed for the
presented model. Finally, some numerical experiments are performed to show
the applications ofthe model and the algorithm.
39
60
Lixing
Yang
Lixing
Yang
State Key Laboratory of Rail Traffic Control and Safety, Beijing
Jiaotong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control
China
lxyang@bjtu.edu.cn
Xiang
Li
Xiang
Li
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control
China
xiangli04@tsinghua.edu.cn
Ziyou
Gao
Ziyou
Gao
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control
China
gaoziyou@jtys.bjtu.edu.cn
Keping
Li
Keping
Li
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control
China
likeping@jtys.bjtu.edu.cn
Minimum risk model
Railway transportation planning
Credibility measure
Discretization method
[bibitem{Crainic2} T. G. Crainic and J. Roy, {it O. R. tools for tactical freight##transportation planning}, European Journal of Operational##Research, textbf{33(3)} (1988), 290297. ##bibitem{Gorman} M. F. Gorman, {it An application of genetic##and tabu searches to the freight railroad operating plan problem},##Annals of Operations Research, textbf{78} (1998), 5169. ##bibitem{Gorman2} M. F. Gorman, {it Santa fe railway uses an operatingplan model to improve its service##design}, Interfaces, textbf{28(4)} (1998), 112. ##bibitem{Keaton} M. H. Keaton, {it Designing optimal railroad operating plans:##lagrangian relaxation and heuristic approaches}, Transportation##Research B, textbf{23(6)} (1989), 415431. ##bibitem{fuzzyexp}B. Liu and Y. K. Liu, {it Expected value of fuzzy variable and fuzzy expected value model}, IEEE##Transactions on Fuzzy Systems, textbf{10(4)} (2002), 445450. ##bibitem{liu2} B. Liu, {it Theory and practice of uncertain programming}, PhysicaVerlag, Heidelberg, 2002. ##bibitem{liu2004} B. Liu, {it Uncertainty theory: an introduction to its axiomatic foundations},##SpringerVerlag, Berlin, 2004. ##bibitem{Liuyk1} Y. K. Liu, {it Convergence results about the use of fuzzy simulation in fuzzy optimization problems}, IEEE##Transactions on Fuzzy Systems, textbf{14} (2006), 295304. ##bibitem{Liuyk2} Y. K. Liu, {it The approximate method for twostagefuzzy random programming with recourse}, IEEE##Transactions on Fuzzy Systems, textbf{15} (2007), 11971208. ##bibitem{Liuyk3} Y. K. Liu, {it The convergence results about approximating fuzzy random minimum risk problems}, Applied## Mathematics and Computation, textbf{205(2)} (2008), 608621. ##bibitem{Nahmias} S. Nahmias, {it Fuzzy variable}, Fuzzy Sets and Systems,##textbf{1} (1978), 97110. ##bibitem{Newton} H. N. Newton, C. Barnhart and P. H. Vance, {it Constructing railroad##blocking plans to minimize handling costs}, Transportation##Science, textbf{32(4)} (1998), 330345. ##bibitem{qin2} Z. Qin and X. Gao, {it Fractional Liu process with application to finance},##Mathematical and Computer Modeling, textbf{50} (2009), 15381543. ##bibitem{qin3} Z. Qin, X. Li and X. Ji, {it Portfolio selection based on fuzzy crossentropy},##Journal of Computational and Applied Mathematics, textbf{228} (2009), 139149. ##bibitem{qin1} Z. Qin and X. Ji, {it Logistics network design for product recovery in fuzzy environment},##European Journal of Operational Research, textbf{202} (2010), 279290. ##bibitem{yang1} L. Yang and L. Liu, {it Fuzzy fixed charge solid transportation problem and algorithm},##Applied Soft Computing, textbf{7} (2007), 879889. ##bibitem{yang2} L. Yang, K. Li and Z. Gao, {it Train timetable problem on a singleline railway with##fuzzy passenger demand}, IEEE Transactions on Fuzzy Systems,##textbf{17(3)} (2009), 617629. ##bibitem{yang3} L. Yang, Z. Gao and K. Li, {it Railway freight transportation planning with mixed##uncertainty of randomness and fuzziness}, Applied Soft##Computing, textbf{11(1)} (2011), 778792 . ##bibitem{yang4} L. Yang, X. Ji, Z. Gao and K. Li, {it Logistics distribution centers##location problem and algorithm under fuzzy environment}, Journal##of Computational and Applied Mathematics, textbf{208(2)} (2007), 303315. ##bibitem{zadeh} L. A. Zadeh, {it Fuzzy sets}, Information and Control,##textbf{8} (1965), 338353. ##bibitem{zhou} J. Zhou and B. Liu, {it Convergence concept of bifuzzy sequence}, Asian InformationScienceLife,##textbf{2(3)} (2004), 297310.##]
MEANABSOLUTE DEVIATION PORTFOLIO SELECTION
MODEL WITH FUZZY RETURNS
MEANABSOLUTE DEVIATION PORTFOLIO SELECTION
MODEL WITH FUZZY RETURNS
2
2
In this paper, we consider portfolio selection problem in which
security returns are regarded as fuzzy variables rather than random variables.
We first introduce a concept of absolute deviation for fuzzy variables and
prove some useful properties, which imply that absolute deviation may be
used to measure risk well. Then we propose two meanabsolute deviation
models by defining risk as absolute deviation to search for optimal portfolios.
Furthermore, we design a hybrid intelligent algorithm by integrating genetic
algorithm and fuzzy simulation to solve the proposed models. Finally, we
illustrate this approach with two numerical examples.
1
In this paper, we consider portfolio selection problem in which
security returns are regarded as fuzzy variables rather than random variables.
We first introduce a concept of absolute deviation for fuzzy variables and
prove some useful properties, which imply that absolute deviation may be
used to measure risk well. Then we propose two meanabsolute deviation
models by defining risk as absolute deviation to search for optimal portfolios.
Furthermore, we design a hybrid intelligent algorithm by integrating genetic
algorithm and fuzzy simulation to solve the proposed models. Finally, we
illustrate this approach with two numerical examples.
61
75
Zhongfeng
Qin
Zhongfeng
Qin
School of Economics and Management, Beihang University, Beijing
100191, China
School of Economics and Management, Beihang
China
qin@buaa.edu.cn, qzf05@mails.thu.edu.cn
Meilin
Wen
Meilin
Wen
School of Reliability and Systems Engineering, Beihang University,
Beijing 100191, China
School of Reliability and Systems Engineering,
China
wenmeilin@buaa.edu.cn
Changchao
Gu
Changchao
Gu
Sinopec Management Institute, Beijing 100012, China
Sinopec Management Institute, Beijing 100012,
China
guchangchao@gamil.com
Uncertainty modelling
Fuzzy variable
Fuzzy portfolio selection
Credibility theory
Hybrid intelligent algorithm
[bibitem{Abdelaziz} F. Abdelaziz, B. Aouni and R. Fayedh, {it##Multiobjective stochastic programming for portfolio selection},##Eur. J. Oper. Res., {bf 177} (2007), 18111823. ##bibitem{ArenasParra} M. ArenasParra, A. BilbaoTerol and M. Rodr'{i}guezUr'{i}a,##{it A fuzzy goal programming approach to portfolio selection}, Eur.##J. Oper. Res., {bf 133} (2001), 287297. ##bibitem{Bilbao} A. BilbaoTerol, B. P'{e}rezGladish, M. ArenasParra and M. Rodr'{i}gezUr'{i}a,##{it Fuzzy compromise programming for portfolio selection}, Appl.##Math. Comput., {bf 173} (2006), 251264. ##bibitem{Corazza} M. Corazza and D. Favaretto, {it On the existence of##solutions to the quadratic mixedinteger meanvariance portfolio##selection problem}, Eur. J. Oper. Res., {bf 176} (2007), 19471980. ##%bibitem{Deng} X. Deng, Z. Li and S. Wang, {it A minimax portfolio##%selection strategy with equilibrium}, Eur. J. Oper. Res., {bf##%166}(2005), 278292. ##%bibitem{Grootveld} H. Grootveld and W. Hallerbach, {it Variance vs##%downside risk: is there really that much difference?}, Eur. J. Oper.##%Res., {bf 114}(1999), 304319. ##%bibitem{huang chance constrained}X. Huang, {it Fuzzy chanceconstrained##%portfolio selection}, Appl. Math. Comput., {bf 177}(2006),##%500507. ##%bibitem{Huang mean variance} X. Huang, {it Portfolio selection with##%fuzzy returns}, J. Intell. Fuzzy Syst., {bf 18}(2007), 383390. ##bibitem{huang mean semivariance} X. Huang, {it Meansemivariance models##for fuzzy protfolio selection}, J. Comput. Appl. Math., {bf##217} (2008), 18. ##bibitem{Konno and Yamazaki} H. Konno and H. Yamazaki, {it Meanabsolute##deviation portfolio optimization model and its applications to Tokyo##Stock Market}, Manage. Sci., {bf 37} (1991), 519531. ##bibitem{li and liu 2006} X. Li and B. Liu, {it A sufficient and necessary condition for credibility##measures}, Int. J. Uncertain. Fuzz., {bf 14} (2006), 527535. ##bibitem{Li and Liu entropy} X. Li and B. Liu, {it Maximum entropy##principle for fuzzy variables}, Int. J. Uncertain. Fuzz., {bf##15} (2007), 4353. ##bibitem{li and qin skewness} X. Li, Z. Qin and S. Kar, {it Meanvarianceskewness##model for portfolio selection with fuzzy returns}, Eur. J. Oper.##Res., {bf 202} (2010), 239247. ##bibitem{Liu book} B. Liu, {it Theory and practice of uncertain##programming}, PhysicaVerlag, Heidelberg, 2002. ##bibitem{liu 2007} B. Liu, {it Uncertainty theory}, 2nd ed.,##SpringerVerlag, Berlin, 2007. ##bibitem{liu iwamura} B. Liu and K. Iwamura, {it Chance constrained##programming with fuzzy parameters}, Fuzzy Sets and Systems, {bf##94} (1998), 227237. ##bibitem{liu and liu 2002} B. Liu and Y. Liu, {it Expected value of fuzzy variable and##fuzzy expected value models}, IEEE T. Fuzzy Syst., {bf 10} (2002),##%bibitem{Liu wang} S. Liu, S. Wang and W. Qiu, {it A##%meanvarianceskewness model for portfolio selection with##%transaction costs}, Int. J. Syst. Sci, {bf 34}(2003), 255262. ##bibitem{ykliu1} Y. Liu, {it Convergent results about the use of fuzzy simulation in fuzzy optimization##problems}, IEEE T. Fuzzy Syst., {bf 14} (2006), 295304. ##bibitem{Markowitz 52} H. Markowitz, {it Porfolio selection}, J. Finance, {bf 7} (1952), 7791. ##bibitem{Markowitz 59} H. Markowitz, {it Portfolio selection: efficient##diversification of investments}, Wiley, New York, 1959. ##%bibitem{Markowitz 90} H. Markowitz, {it Computation of##%meansemivariance efficient sets by the critical line algorithm},##%Ann. Oper. Res., {bf 45}(1993), 307317. ##bibitem{Qin Li Ji} Z. Qin, X. Li and X. Ji, {it Portfolio selection based on##fuzzy crossentropy}, J. Comput. Appl. Math., {bf 228} (2009),##%bibitem{Roy} A.D. Roy, {it Safety first and the holding of assets}, Econometrics, {bf 20}(1952), 431449. ##bibitem{Simaan} Y. Simaan, {it Estimation risk in portfolio selection:##the mean vairance model versus the mean absolute deviation model},##Manage. Sci., {bf 43} (1997), 14371446. ##bibitem{Speranza} M. G. Speranza, {it Linear programming model for##portfolio optimization}, Finance, {bf 14} (1993), 107123. ##bibitem{Tanaka guo} H. Tanaka and P. Guo, {it Portfolio selection based on##upper and lower exponential possibility distributions}, Eur. J.##Oper. Res., {bf 114} (1999), 115126. ##bibitem{Tanaka guo Turksen} H. Tanaka, P. Guo and I. T"{u}rksen, {it Portfolio selection based on##fuzzy probabilities and possibility distributions}, Fuzzy Sets and##Systems, {bf 111} (2000), 387397. ##bibitem{Vercher} E. Vercher, J. Berm'{u}dez and J. Segura, {it Fuzzy##portfolio optimization under downside risk measures}, Fuzzy Sets and##Systems, {bf 158} (2007), 769782. ##bibitem{Yang 1} L. Yang, K. Li and Z. Gao, {it Train timetable##problem on a singleline railway with fuzzy passenger demand}, IEEE##T. Fuzzy Syst., {bf 17} (2009), 617629. ##bibitem{Yang 2} L. Yang and L. Liu, {it Fuzzy fixed charge solid transportation##problem and algorithm}, Appl. Soft Comput., {bf 7} (2007), 879889. ##bibitem{Yang 3} L. Yang, Z. Gao and K. Li, {it Railway##freight transportation planning with mixed uncertainty of randomness##and fuzziness}, Appl. Soft Comput., {bf 11} (2011), 778792. ##bibitem{Zadeh} L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf 8} (1965), 338353. ##bibitem{Zhang} W. Zhang and Z. Nie, {it On admissible efficient##portfolio selection problem}, Appl. Math. Comput., {bf 159} (2004),##]
FUZZY TRAIN ENERGY CONSUMPTION MINIMIZATION
MODEL AND ALGORITHM
FUZZY TRAIN ENERGY CONSUMPTION MINIMIZATION
MODEL AND ALGORITHM
2
2
Train energy saving problem investigates how to control train's
velocity such that the quantity of energy consumption is minimized and some
system constraints are satis ed. On the assumption that the train's weights
on different links are estimated by fuzzy variables when making the train
scheduling strategy, we study the fuzzy train energy saving problem. First, we
propose a fuzzy energy consumption minimization model, which minimizes the
average value and entropy of the fuzzy energy consumption under the maximal
allowable velocity constraint and traversing time constraint. Furthermore, we
analyze the properties of the optimal solution, and then design an iterative
algorithm based on the KarushKuhnTucker conditions. Finally, we illustrate
a numerical example to show the effectiveness of the proposed model and
algorithm.
1
Train energy saving problem investigates how to control train's
velocity such that the quantity of energy consumption is minimized and some
system constraints are satis ed. On the assumption that the train's weights
on different links are estimated by fuzzy variables when making the train
scheduling strategy, we study the fuzzy train energy saving problem. First, we
propose a fuzzy energy consumption minimization model, which minimizes the
average value and entropy of the fuzzy energy consumption under the maximal
allowable velocity constraint and traversing time constraint. Furthermore, we
analyze the properties of the optimal solution, and then design an iterative
algorithm based on the KarushKuhnTucker conditions. Finally, we illustrate
a numerical example to show the effectiveness of the proposed model and
algorithm.
77
91
Xiang
Li
Xiang
Li
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao
tong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control
China
xiangli04@mail.tsinghua.edu.cn
Dan
Ralescu
Dan
Ralescu
Department of Mathematical Sciences, University of Cincinnati, Cincin
nati, Ohio 45221, USA
Department of Mathematical Sciences, University
United States
ralescd@uc.edu
Tao
Tang
Tao
Tang
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao
tong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control
China
ttang@bjtu.edu.cn
Energy consumption
Train scheduling
KarushKuhnTucker condi tions
Fuzzy variable
[[1] Y. Bai, B. Mao, F. Zhou, Y. Ding and C. Dong,Energyecient driving strategy for freight trains based on power consumption analysis, Journal of Transportation Systems Engineering and Information Technology,9(3) (2009), 4350. ##[2] E. Castillo, I. Gallego, J. M. Urena and J. M. Coronado,Timetabling optimization of a single railway track line with sensitivity analysis, Top, 17(2) (2009), 256287. ##[3] C. S. Chang and S. Sim,Optimising train movements through coast control using genetic algorithms, IEE ProceedingsElectric Power Applications, 144(1) (1997), 6573. ##[4] S. Eati and H. Roohparvar,The minimization of the fuel costs in the train transportation,Applied Mathematics and Computation,175 (2006), 14151431. ##[5] K. Ghoseiri, F. Szidarovszky and M. J. Asgharpour,Amultiobjective train scheduling model and solution, Transportation Research Part B, 38 (2004), 927952. ##[6] P. Howlett,The optimal control of a train, Annals of Operations Research, 98 (2000), 6587. ##[7] P. Howlett, P. Pudney and X. Vu,Local energy minimization in optimal train control, Automatica,45(11)(2009), 26922698. ##[8] K. B. Khan and X. S. Zhou,Stochastic optimization model and solution algorithm for robust doubletrack traintimetabling problem, IEEE Transactions on Intelligent Transportation Systems,11(1)(2010), 8189. ##[9] E. Khmelnitsky,On an optimal control problem of train operation, IEEE Transactions on Automatic Control,45(7) (2000), 12571266. ##[10] D. R. Kraay, P. T. Harker and B. Chen,Optimal pacing of trains in freight railroads: model formulation and solution, Operations Research, 39 (1991), 8299. ##[11] X. Li and B. Liu,A sucient and necessary condition for credibility measure, Internationa Journal of Uncertainty, Fuzziness & KnowledgeBased Systems,14(5) (2006), 527535. ##[12] P. Li and B. Liu,Entropy of credibility distributions for fuzzy variables, IEEE Transactions on Fuzzy Systems,16(1) (2008), 123129. ##[13] B. Liu and Y. Liu,Expected value of fuzzy variable and fuzzy expected value models, IEEETransactions on Fuzzy Systems,10(4) (2002), 445450. ##[14] R. Liu and I. M. Golovitcher,Energyecient operation of rail vehicles, Transportation Research Part A,37(2003), 917932. ##[15] M. Miyatake and H. Ko,Optimization of train speed pro le for minimum energy consumption,IEEJ Transactions on Electrical and Electronic Engineering,5 (2010), 263269. ##[16] L. Yang, K. P. Li and Z. Y. Gao,Train timetable problem on a singleline railway with fuzzy passenger demand, IEEE Transactions on Fuzzy Systems, 17(3)(2009), 617629. ##[17] L. Yang, Z. Y. Gao and K. P. Li,Passenger train scheduling on a singletrack or partially doubletrack railway with stochastic information, Engineering Optimization, 42(11) (2010),10031022.##]
FLUENCE MAP OPTIMIZATION IN INTENSITY MODULATED
RADIATION THERAPY FOR FUZZY TARGET DOSE
FLUENCE MAP OPTIMIZATION IN INTENSITY MODULATED
RADIATION THERAPY FOR FUZZY TARGET DOSE
2
2
Although many methods exist for intensity modulated radiotherapy (IMRT) fluence map optimization for crisp data, based on clinical practice, some of the involved parameters are fuzzy. In this paper, the best fluence maps for an IMRT procedure were identifed as a solution of an optimization problem with a quadratic objective function, where the prescribed target dose vector was fuzzy. First, a defuzzyingprocedure was introduced to change the fuzzy model of the problem into an equivalent nonfuzzy one. Since the solution set was nonconvex, the optimal solution was then obtained by performing a projection operation in applying the gradient method. Numerical simulations for two typical clinical cases (for prostate and headandneck cancers, each for two patients) are given.
1
Although many methods exist for intensity modulated radiotherapy (IMRT) fluence map optimization for crisp data, based on clinical practice, some of the involved parameters are fuzzy. In this paper, the best fluence maps for an IMRT procedure were identifed as a solution of an optimization problem with a quadratic objective function, where the prescribed target dose vector was fuzzy. First, a defuzzyingprocedure was introduced to change the fuzzy model of the problem into an equivalent nonfuzzy one. Since the solution set was nonconvex, the optimal solution was then obtained by performing a projection operation in applying the gradient method. Numerical simulations for two typical clinical cases (for prostate and headandneck cancers, each for two patients) are given.
93
105
Alireza
Fakharzadeh Jahrom
Alireza
Fakharzadeh Jahromi
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz,
Iran
a_ fakharzadeh@sutech.ac.ir
Omolbanin
Bozorg
Omolbanin
Bozorg
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz,
Iran
o.bozorg@gmail.com
Hamidreza
Maleki
Hamidreza
Maleki
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz,
Iran
maleki@sutech.ac.ir
Mohamad Amin
MoslehShirazi
Mohamad Amin
MoslehShirazi
Shiraz University of Medical Sciences, Shiraz, Fars,
Shiraz University of Medical Sciences, Shiraz,
Iran
mosleh_amin@hotmail.com
IMRT
Singed distance
Triangular fuzzy number
Gradient method
[bibitem{2} G. Bahr, J. Kereiakes, H. Horwitz, R. Finnery, J. Calvin and##K. Goode, {it The method of linear programming applied to##radiation treatment planning}, Radiology, {bf 91} (1968), 686693. ##bibitem{salmani ref1} J. Chiang, {it Fuzzy linear programming based on statistical confidence interval and## intervalvalued fuzzy set}, European Journal of##Operational Research, {bf 129} (2001), 6586. ##bibitem{8} C. Cotrutz, M. Lahanas, C. Kappas and D. Baltas, {it A##multiobjective gradient based dose optimization algorithm for##external beam conformal radiotherapy}, Physics in Medicine and Biology, {bf 46} (2001), 21612175. ##bibitem{CERR} J. Deasy, A. Blanco and V. Clark, {it Cerr: a computational environment for radiotherapy research}, Medical Physics, {bf 30} (2003), 979985. ##bibitem{Dubois} D. Dubois and H. Prade, {it Fuzzy sets and systems: theory and application},## Academic Press, New York, 1980. ##bibitem{Hamacher} H. W. Hamacher and K. H. Kufer, {it Inverse radiation therapy planninng  a multiple objective optimization##approach}, Discrete Applied Mathematics, {bf 118} (2002), 145161. ##bibitem{15} C. L. Lawson and R. J. Hanson, {it Solving least squares problems}, SIAM, (1995), 160165. ##bibitem{mixed integer} E. Lee, T. Fox and I. Crocker, {it Integer programming applied to intensity##modulated radiation therapy treatment planning}, Annals of##Operations Research, {bf 119} (2003), 165181. ##bibitem{IMRT1} M. S. Merritt, {it A sensitivitydriven greedy approach to fluence map##optimization in intensitymodulated radiation therapy}, Ph. D. thesis, Department of Computational and Applied Mathematics,##Rice University, April 2006. ##bibitem{26} S. Morrill, R. Lane, J. Wong and I. Rosen, {it Dosevolume##considerations with linear programming optimization}, Medical Physics, {bf 18} (1991), 12011210. ##bibitem{Preciado} F. PreciadoWalters, M. Langer, R. Rardin and V. Thai, {it A coupled column generation, mixedinteger##approach to optimal planning of intensity modulated therapy for##cancer}, Mathematical Programming, {bf 101} (2004), 319338. ##bibitem{Romeijn} H. E. Romeijn, R. K. Ahuja, J. F. Dempsey, A. Kumar and G. J. Li, {it Novel linear programming approach##to fluence map optimization for intensity modulated radiation##therapy treatment planning}, Physics in Medicine and Biology, {bf 48} (2003), 35213542. ##bibitem{30} H. E. Romeijn, J. F. Dempsey and J. G. Li, {it A unifying framework for multi##criteria fluence map optimization models}, Physics in Medicine and Biology, {bf 49} (2004), 19912013. ##bibitem{34} S. V. Spirou and C. Chui, {it A gradient inverse planning algorithm with dosevolume constraints}, Medical Physics, {bf 25} (1998), 321333. ##bibitem{Yao wu} J. Yao and K. Wu, {it Ranking fuzzy numbers based on decomposition principle and signed##distance}, Fuzzy Sets and Systems, {bf 116} (2000), 275288. ##bibitem{Yin Zhang} Y. Zhang and M. Merrit, {it Dosevolumebased IMRT fluence optimization: a fast##leastsquares approach with differentiability}, Linear Algebra and##its Applications, {bf 428} (2008), 13651387. ##bibitem{zimerman} H. J. Zimmermann, {it Fuzzy set theory and its applications},## second ed., Kluwer Academic Publishers, Boston, 1991.##]
COALITIONAL GAME WITH FUZZY PAYOFFS AND
CREDIBILISTIC SHAPLEY VALUE
COALITIONAL GAME WITH FUZZY PAYOFFS AND
CREDIBILISTIC SHAPLEY VALUE
2
2
Coalitional game deals with situations that involve cooperations among players, and there are different solution concepts such as the core,the Shapley value and the kernel. In many situations, there is no way to predict the payoff functions except for the expert experiencesand subjective intuitions, which leads to the coalitional game with fuzzy payoffs. Within the framework of credibility theory, this paper employstwo credibilistic approaches to define the behaviors of players under fuzzy situations. Correspondingly, two variations of Shapley value areproposed as the solutions of the coalitional game with fuzzy payoffs. Meanwhile, some characterizations of the credibilistic Shapley valueare investigated. Finally, an example is provided for illustrating the usefulness of the theory developed in this paper.
1
Coalitional game deals with situations that involve cooperations among players, and there are different solution concepts such as the core,the Shapley value and the kernel. In many situations, there is no way to predict the payoff functions except for the expert experiencesand subjective intuitions, which leads to the coalitional game with fuzzy payoffs. Within the framework of credibility theory, this paper employstwo credibilistic approaches to define the behaviors of players under fuzzy situations. Correspondingly, two variations of Shapley value areproposed as the solutions of the coalitional game with fuzzy payoffs. Meanwhile, some characterizations of the credibilistic Shapley valueare investigated. Finally, an example is provided for illustrating the usefulness of the theory developed in this paper.
107
117
Jinwu
Gao
Jinwu
Gao
Uncertain Systems Lab, School of Information, Renmin University of
China, Beijing 100872, China
Uncertain Systems Lab, School of Information,
China
jgao@ruc.edu.cn
Q.
Zhang
Q.
Zhang
Uncertain Systems Lab, School of Information, Renmin University of
China, Beijing 100872, China
Uncertain Systems Lab, School of Information,
China
zqw2002@163.com
P.
Shen
P.
Shen
Uncertain Systems Lab, School of Information, Renmin University of China,
Beijing 100872, China
Uncertain Systems Lab, School of Information,
China
shenpuchen@163.com
Coalitional game
Shapley value
Fuzzy variable
Credibility measure
[bibitem{Aumann64} R. Aumann and M. Maschler, {it The bargainning set##for cooperative games}, Advances in Game Theory, Annals of##Mathematics Studies, {bf 52} (1964), 443476. ##bibitem{Aumann74} R. Aumann and L. S. Shapley, {it Values of nonatomic games}, Princeton University,##Princeton, 1974. ##bibitem{Aubin81} J. P. Aubin, {it Cooperative fuzzy games}, Mathematical Operation##Research, {bf 6} (1981), 113. ##bibitem{Butnariu78} D. Butnariu, {it Fuzzy games: a description of the##concept}, Fuzzy Sets and Systems, {bf 1} (1978), 181192. ##bibitem{Butnariu80} D. Butnariu, {it Stability and shapley value for an npersons fuzzy games}, Fuzzy Sets and Systems, {bf 4} (1980), 6372. ##bibitem{Butnariu96} D. Butnariu and E. P. Klement, {it Core, value and equilibria for market games: on a problem of Aumann and Shapley}, International Journal of Game Theory, {bf 18} (1996), 149160. ##bibitem{Campos92} L. Campos, A. Gonzalez and M. A. Vila, {it On the use of the ranking function approach to solve fuzzy##matrix games in a direct way}, Fuzzy Sets and Systems, {bf 49} (1992), 193203. ##bibitem{dubios88} D.~Dubois and H.~Prade, {it Possibility theory},##Plenum, New York, 1988. ##bibitem{Gao05} J.~Gao, and B.~Liu, {it Fuzzy multilevel programming with a hybrid intelligent##algorithm}, Computer & Mathmatics with applications, {bf 49} (2005), 15391548. ##bibitem{Gao07} J. Gao, {it Credibilistic game with fuzzy##information}, Journal of Uncertain Systems, {bf 1(1)} (2007), 7278. ##bibitem{Gao09} J. Gao, Z. Q. Liu and P. Shen, {it On characterization of fuzzypayoff twoplayer zerosum game}, Soft Computing, {bf 13(2)} (2009), 127132.##bibitem{Gao111} J. Gao, {it Fuzzy dependentchance multilevel programming with applications}, Journal of Universal Computer Science, to be Published.##bibitem{Gao112} J. Gao and Y. Yu, {it Credibilistic extensive game with fuzzy information}, Soft Computing, to be Published. ##bibitem{Harsanyi95} J. C. Harsanyi, {it Games with incomplete information},## The American Economic Review, {bf 85(3)} (1995), 291303. ##bibitem{Gao101} R. Liang, Y. Yu, J. Gao and Z. Q. Liu, {it Nperson credibilistic strategic game}, Frontiers of Computer Science in China, {bf 4(2)} (2010), 212219.##bibitem{Liu02} B.~Liu and Y.~Liu, {it Expected value of fuzzy variable and fuzzy##expected value models}, IEEE Transactions on Fuzzy Systems, {bf 10} (2002), 445450. ##bibitem{Liu03} Y. Liu and B. Liu, {it Expected value operator of random fuzzy variable and random##fuzzy expected value models}, International Journal of##Uncertainty, Fuzziness & KnowledgeBased Systems, {bf 11(2)} (2003), 195215. ##bibitem{Liubook04} B.~Liu, {it Uncertainty theory: an introduction to its axiomatic##foundations}, SpringerVerlag, Berlin, 2004. ##bibitem{Liu06} B. Liu, {it A survey of credibility theory}, Fuzzy Optimization and Decision Making,##{bf 5(4)} (2006), 387408. ##bibitem{Liu07} Y. Liu and J. Gao, {it The dependence of fuzzy variables with applications##to fuzzy random optimization}, International Journal of##Uncertainty, Fuzziness & KnowledgeBased Systems, {bf 15} (2007), 120. ##bibitem{Liubook07} B.~Liu, {it Uncertainty theory}, 2nd Edition, SpringerVerlag, Berlin, 2007. ##bibitem{Neumann44} J. Von Neumann and D. Morgenstern, {it The theory of games in economic bahavior},##New York: Wiley, 1944. ##bibitem{Maeda00} T. Maeda, {it Characterization of the equilibrium strategy of the bimatrix##game with fuzzy payoff}, Journal of Mathematical Analysis and##Applications, {bf 251} (2000), 885896. ##bibitem{Maeda03} T. Maeda, {it Characterization of the equilibrium strategy of the twoperson##zerosum game with fuzzy payoff}, Fuzzy Sets and Systems, {bf 139} (2003), 283296. ##bibitem{Mares94} M. Mareu{s}, {it Computation over fuzzy quantities}, CRCPress, 1994. ##bibitem{Mares00} M. Mareu{s}, {it Fuzzy coalitions structures}, Fuzzy Sets##and Systems, {bf 114(1)} (2000), 2333. ##bibitem{Mares01} M. Mareu{s}, {it Fuzzy cooperative games}, PhysicaVerlag, Heidelberg, 2001. ##bibitem{Maschler79} M. Mashler, B. Peleg and L. S. Shapley, {it Geometric##properties of the kernel, nucleolus and related solution concepts},## Mathematics Operational Research, {bf 4} (1979), 303337. ##bibitem{Nishizaki00} I. Nishizaki and M. Sakawa, {it Equilibrium solutions for multiobjective bimatrix games with fuzzy payoffs##and fuzzy goals}, Fuzzy Sets and Systems, {bf 111(1)} (2000), 99116. ##bibitem{Nishizaki00a} I. Nishiizaki and M. Sakawa, {it Fuzzy cooperative games##arising from linear production programming problems with fuzzy##parameters}, Fuzzy sets and Systems, {bf 114(1)} (2000), 1121. ##bibitem{Nishizaki00b} I. Nishiizaki and M. Sakawa, {it Solutions based##on fuzzy goals in fuzzy linear programming games}, Fuzzy Sets##and Systems, {bf 115(1)} (2000), 105109. ##bibitem{Shapley53} L. S. Shapley, {it A value for npersons games}, Annals of Mathematics Studies, {bf 28} (1953), 307318. ##bibitem{Shapley54} L. S. Shapley and M. Shubik, {it A method for evaluating##the distribution of power in committee systerm}, American##Politics Science Review, {bf 48} (1954), 787792. ##bibitem{Schmeidler69} D. Schmeidler, {it The nucleolus of a##characteristic function games}, Journal of Applied Mathematics##{bf 17} (1969), 11631170. ##bibitem{Gao102} P. Shen and J. Gao, {it Coalitional game with fuzzy information and credibilistic core}, Soft Computing, DOI: 10.1007/s0050001006329, 2010. ##%bibitem{Peleg03} B. Peleg, {em Introdutin to the theory of cooperative##%game}, Kluswer Academic Publishers, Boston, 2003. ##bibitem{zadeh65} L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf 8}(1965), 338353. ##bibitem{zadeh78} L. A. Zadeh, {it Fuzzy sets as a basis for a theory of##possibility}, Fuzzy Sets and Systems, {bf 1}(1978), 328.vspace{.3 cm}##]
Persiantranslation vol. 8, no. 4, October 2011
2
2
1

121
127