ORIGINAL_ARTICLE
Cover Special Issue vol. 8, no. 4, October 2011
http://ijfs.usb.ac.ir/article_2868_3f83c7d9389a4164783e5336fe598543.pdf
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10.22111/ijfs.2011.2868
ORIGINAL_ARTICLE
SOME PROPERTIES FOR FUZZY CHANCE CONSTRAINED
PROGRAMMING
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.
http://ijfs.usb.ac.ir/article_305_6fb6b2d1b824c72a0c944ba1a2867828.pdf
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8
10.22111/ijfs.2011.305
Convexity theorem
Duality theorem
Fuzzy variable
Chance con-
strained programming
Xiaohu
Yang
yxh12@163.com
true
1
Department of Statistics, Xi'an University of Finance and Economics,
Xi'an 710061, China
Department of Statistics, Xi'an University of Finance and Economics,
Xi'an 710061, China
Department of Statistics, Xi'an University of Finance and Economics,
Xi'an 710061, China
LEAD_AUTHOR
bibitem{Bellman and Zadeh}R. E. Bellman and L. A. Zadeh, {it Decision making in a fuzzy environment},
1
Management Science, {bf textbf{17}} (1970), 141-164.
2
bibitem{Delgado}M. Delgado, J. Kacprzyk, J. L. Verdegay and M. A. Vila, {it Fuzzy optimization: recent advances},
3
Physica-Verlag, Wurzburg, 1994.
4
bibitem{Ji}X. Ji X and Z. Shao, {it Model and algorithm for bilevel newsboy
5
problem with fuzzy demands and discounts}, Applied Mathematics and
6
Computation, {bf textbf{172(1)}} (2006), 163-174.
7
bibitem{Lai 1}Y. J. Lai and C. L. Hwang, {it Fuzzy mathematical programming: methods and applications},
8
Lecture Notes in Economics and Mathematical Systems,
9
Springer, Berlin, {bftextbf{394}} (1992).
10
bibitem{Lai 2}Y. J. Lai and C. L. Hwang, {it Fuzzy multiple objective decision making: methods and applications},
11
Lecture notes in Economics and Mathematical Systems,
12
Springer, Berlin, {bftextbf{404}} (1992).
13
bibitem{Li 2006}X. Li and B. Liu, {it A sufficient and necessary condition for credibility measures},
14
International Journal of Uncertainty, Fuzziness & Knowledge-Based
15
Systems, {bf textbf{14(5)}} (2006), 527-535.
16
bibitem{Li and liu}X. Li and B. Liu, {it The independence of fuzzy variables with
17
applications}, International Journal of Natural Sciences and
18
Technology, {bf textbf{1(1)}} (2006), 95-100.
19
bibitem{Li2010}X. Li, Z. Qin and L. X. Yang, {it A chance-constrained portfolio selection model with risk
20
constraints}, Applied Mathematics and Computation, {bf
21
textbf{217}} (2010), 949-951.
22
bibitem{Li2011}X. Li, Z. Qin, L. X. Yang and K. P. Li, {it Entropy maximization model for trip distribution problem with
23
fuzzy and random parameters}, Journal of Computational and Applied
24
Mathematics, doi:10.1016/j.cam.2010.09.004.
25
bibitem{Liu maximax ccp}B. Liu and K. Iwamura, {it Chance constrained programming with fuzzy parameters},
26
Fuzzy Sets and Systems, {bf textbf{94(2)}} (1998), 227-237.
27
bibitem{Liu minimax ccp}B. Liu, {it Minimax chance constrained programming models for fuzzy decision systems},
28
Information Sciences, {bf textbf{112(1-4)}} (1998), 25-38.
29
bibitem{Liu dcp}B. Liu, {it Dependent-chance programming in fuzzy environment}, Fuzzy Sets and Systems, {bf textbf{109(1)}} (2000), 97-106.
30
bibitem{Liu 2002}B. Liu, {em Theory and practice of uncertain programming}, Physica-Verlag, Heidelberg, 2002.
31
bibitem{Liu and Liu Evofv}B. Liu and Y. K. Liu, {it Expected value of fuzzy variable and fuzzy expected value models},
32
IEEE Transactions on Fuzzy Systems, {bf textbf{10(4)}} (2002),
33
bibitem{Liu 2004}B. Liu, {em Uncertainty theory: an introduction to its axiomatic
34
foundations}, Springer-Verlag, Berlin, 2004.
35
bibitem{Liu 2007}B. Liu, {em Uncertainty theory}, 2nd ed.,
36
Springer-Verlag, Berlin, 2007.
37
bibitem{Liu and Gao independent}Y. K. Liu and J. Gao, {it The independence of fuzzy variables with applications to fuzzy random
38
optimization}, International Journal of Uncertainty, Fuzziness &
39
Knowledge-Based Systems, {bf textbf{15(Suppl.2)}} (2007), 1-20.
40
bibitem{peng and liu}J. Peng and B. Liu, {it Parallel machine scheduling models with fuzzy processing times},
41
Information Sciences, {bf textbf{166(1-4)}} (2004), 49-66.
42
bibitem{slowinski}R. Slowinski, {em Fuzzy sets in decision analysis,
43
operations research and statistics}, Kluwer Academic Publishers,
44
Dordrecht, 1998.
45
bibitem{vajda}S. Vajda, {em Probabilistic programming}, Academic Press,
46
New York, 1972.
47
bibitem{Zadeh 1978}L. A. Zadeh, {it Fuzzy sets as a basis for a theory of possibility}, Fuzzy Sets and
48
Systems, {bf textbf{1}} (1978), 3-28.
49
bibitem{Zadeh 1979}L. A. Zadeh, {it A theory of approximate reasoning}, In: Hayes J et al.,
50
Mathematical Frontiers of the Social and Policy Sciences, Westview
51
Press, Boulder, Cororado, (1979), 69-129.
52
bibitem{zhao and liu}R. Zhao and B. Liu, {it Standby redundancy optimization problems with fuzzy lifetimes},
53
Computers & Industrial Engineering, {bf textbf{149(2)}} (2005),
54
bibitem{Zheng and Liu}Y. Zheng and B. Liu, {it Fuzzy vehicle routing model with
55
credibility measure and its hybrid intelligent algorithm}, Applied
56
Mathematics and Computation, {bf textbf{176(2)}} (2006), 673-683.
57
bibitem{Zhou}J. Zhou and B. Liu B, {it Modeling capacitated location-allocation
58
problem with fuzzy demands}, Computers & Industrial Engineering,
59
{bf textbf{53(3)}} (2007), 454-468.
60
ORIGINAL_ARTICLE
ALGORITHMS FOR BIOBJECTIVE SHORTEST PATH
PROBLEMS IN FUZZY NETWORKS
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.
http://ijfs.usb.ac.ir/article_306_645249d97ea17b67c4eb5772a11928ac.pdf
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37
10.22111/ijfs.2011.306
Biobjective shortest path
Fuzzy network
Labeling method
Dynamic
programming
Fuzzy ranking methods
Iraj
Mahdavi
irajarash@rediffmail.com
true
1
Department of Industrial Engineering, Mazandaran University of Sci-
ence & Technology, Babol, Iran
Department of Industrial Engineering, Mazandaran University of Sci-
ence & Technology, Babol, Iran
Department of Industrial Engineering, Mazandaran University of Sci-
ence & Technology, Babol, Iran
AUTHOR
Nezam
Mahdavi-Amiri
nezamm@sharif.edu
true
2
Faculty of Mathematical Sciences, Sharif University of Tech-
nology, Tehran, Iran
Faculty of Mathematical Sciences, Sharif University of Tech-
nology, Tehran, Iran
Faculty of Mathematical Sciences, Sharif University of Tech-
nology, Tehran, Iran
AUTHOR
Shahrbanoo
Nejati
nejati sh@yahoo.com
true
3
Department of Industrial Engineering, Mazandaran University
of Science & Technology, Babol, Iran
Department of Industrial Engineering, Mazandaran University
of Science & Technology, Babol, Iran
Department of Industrial Engineering, Mazandaran University
of Science & Technology, Babol, Iran
LEAD_AUTHOR
bibitem{baas1977rating}
1
S. M. Baas and H.~Kwakernaak, {it Rating and ranking of multiple-aspect alternatives using fuzzy
2
sets}, Automatica, {bf 13}textbf{(1)} (1977), 47-58.
3
bibitem{baldwin1979comparison}
4
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), 213-231.
5
bibitem{bellman1958routing}
6
R.~Bellman, {it On a routing problem, Quart}, J. Appl. Math, {bf 16}textbf{(1)} (1958), 87-90.
7
bibitem{bortolan1985review}
8
G.~Bortolan and R.~Degani, {it A review of some methods for ranking fuzzy subsets}, Fuzzy Sets and Systems, {bf 15}textbf{(1)} (1985), 1-19.
9
bibitem{brumbaugh1989empirical}
10
D.~Brumbaugh-Smith and D. Shier, {it An empirical investigation of some bicriterion shortest path
11
algorithms}, European Journal of Operational Research, {bf 43}textbf{(2)} (1989), 216-224.
12
bibitem{campos1989subjective}
13
L.~Campos and A.~Munoz, {it A subjective approach for ranking fuzzy numbers}, Fuzzy Sets and Systems, {bf 29}textbf{(12)} (1989), 145-153.
14
bibitem{chanas1995fuzzy}
15
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), 568-580.
16
bibitem{chang1981ranking}
17
W.~Chang, {it Ranking of fuzzy utilities with triangular membership functions}, In Proc. Int. Conf. of Policy Anal. and Inf. Systems, (1981), 263-272.
18
bibitem{chen1985ranking}
19
S. H. Chen, {it Ranking fuzzy numbers with maximizing set and minimizing set}, Fuzzy Sets and Systems, {bf 17}textbf{(2)} (1985), 113-129.
20
bibitem{choobineh1993index}
21
F.~Choobineh and H.~Li, {it An index for ordering fuzzy numbers}, Fuzzy Sets and Systems, {bf 54}textbf{(3)} (1993), 287-294.
22
bibitem{delgado1990valuation}
23
M.~Delgado, J. L. Verdegay and M. A. Vila, {it On valuation and optimization problems in fuzzy graphs: a general
24
approach and some particular cases}, INFORMS Journal on Computing, {bf 2}textbf{(1)} (1990), 74.
25
bibitem{dijkstra1959note}
26
E. W. Dijkstra, {it A note on two problems in connexion with graphs}, Numerische mathematik, {bf 1}textbf{(1)} (1959), 269-271.
27
bibitem{dreyfus1969appraisal}
28
S. E. Dreyfus, {it An appraisal of some shortest-path algorithms}, Operations Research, {bf 17}textbf{(3)} (1969), 395-412.
29
bibitem{dubois1978algorithmes}
30
D.~Dubois and H.~Prade, {it Algorithmes de plus courts chemins pour traiter des donnees floues.
31
RAIRO-Recherche Op{'e}rationnelle}, Operations Research, {bf 12}(1978), 212--227.
32
bibitem{dubois1980fuzzy}
33
D.~Dubois and H.~Prade, {it Fuzzy sets and systems: theory and applications}, Academic Pr, 1980.
34
bibitem{furukawa1994parametric}
35
N.~Furukawa, {it A parametric total order on fuzzy numbers and a fuzzy shortest route
36
problem}, Optimization, {bf 30}textbf{(4)} (1994), 367-377.
37
bibitem{hansen1980bicriterion}
38
P.~Hansen, {it Bicriterion path problems}, In Multiple criteria decision making: theory and application:
39
proceedings of the third conference, Hagen/K{`e}onigswinter, West Germany,
40
August 20-24, (1979), 109, Springer, 1980.
41
bibitem{helgason1995primal}
42
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), 85-133.
43
bibitem{huarng1996computational}
44
F.~Huarng, P. Pulat, and L. S. Shih, {it A computational comparison of some bicriterion shortest path
45
algorithms}, Journal of the Chinese Institute of Industrial Engineers, {bf 13}textbf{(2)} (1996), 121-125.
46
bibitem{klein1991fuzzy}
47
C. M. Klein, {it Fuzzy shortest paths}, Fuzzy Sets and Systems, {bf 39}textbf{(1)} (1991), 27-41.
48
bibitem{k—czy1992fuzzy}
49
L. T. K{'o}czy, {it Fuzzy graphs in the evaluation and optimization of networks}, Fuzzy Sets and Systems, {bf 46}textbf{(3)} (1992), 307-319.
50
bibitem{lin1993fuzzy}
51
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), 343-353.
52
bibitem{liou1992ranking}
53
T. S. Liou and M. J. J. Wang, {it Ranking fuzzy numbers with integral value}, Fuzzy Sets and Systems, {bf 50}textbf{(3)} (1992), 247-255.
54
bibitem{mahdavi2009dynamic}
55
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), 503-511.
56
bibitem{martins1984multicriteria}
57
E. Q. V. Martins, {it On a multicriteria shortest path problem}, European Journal of Operational Research, {bf 16}textbf{(2)} (1984), 236-245.
58
bibitem{ishwar1991parametric}
59
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), 81-92.
60
bibitem{namorado1982bicriterion}
61
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), 399-404.
62
bibitem{okada2000shortest}
63
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), 129-140.
64
bibitem{prade1979using}
65
H.~Prade, {it Using fuzzy set theory in a scheduling problem: a case study}, Fuzzy Sets and Systems, {bf 2}textbf{(2)} (1979), 153-165.
66
bibitem{przybylski2008two}
67
A.~Przybylski, X.~Gandibleux and M.~Ehrgott, {it Two phase algorithms for the bi-objective assignment problem}, European Journal of Operational Research, {bf 185}textbf{(2)} (2008), 509-533.
68
bibitem{ram’k1985inequality}
69
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), 123-138.
70
bibitem{saade1992ordering}
71
J. J. Saade and H.~Schwarzlander, {it Ordering fuzzy sets over the real line: an approach based on
72
decision making under uncertainty}, Fuzzy Sets and Systems, {bf 50}textbf{(3)} (1992), 237-246.
73
bibitem{sadeghpour2001distance}
74
B.~Sadeghpour~Gildeh and D.~Gien, {it La distance-Dp, q et le cofficient de corr{'e}lation entre deux
75
variables al{'e}atoires floues}, Actes de LFA, (2001), 97-102.
76
bibitem{serafini1987some}
77
P.~Serafini, {it Some considerations about computational complexity for multi
78
objective combinatorial problems}, In Recent Advances and Historical Development of Vector
79
Optimization: Proceedings of an International Conference of Vector
80
Optimization Held at the Technical University of Darmstadt, FRG, August 4-7,
81
(1986), 222, Springer, 1987.
82
bibitem{skriver2000label}
83
A. J. V. Skriver and K. A. Andersen, {it A label correcting approach for solving bicriterion shortest-path problems}, Computers & Operations Research, {bf 27}textbf{(6)} (2000), 507-524.
84
bibitem{skriver2000classification}
85
A. J. V. Skriver, {it A classification of bicriterion shortest path (BSP) algorithms}, Asia Pacific Journal of Operational Research, {bf 17}textbf{(2)} (2000), 199-212.
86
bibitem{TajdinMahdavi2010}
87
A.~Tajdin, I.~Mahdavi, N.~Mahdavi-Amiri and B.~Sadeghpour-Gildeh, {it Computing a fuzzy shortest path in a network with mixed fuzzy arc lengths using $alpha$ -cuts}, Computer and Mathematics with Applications, 2010.
88
bibitem{tung1988bicriterion}
89
C. T. Tung and K. L. Chew, {it A bicriterion Pareto-optimal path algorithm}, ASIA-PACIFIC J. OPER. RES., {bf 5}textbf{(2)} (1988), 166-172.
90
bibitem{tung1992multicriteria}
91
C.~Tung and K.~Lin~Chew, {it A multicriteria Pareto-optimal path algorithm}, European Journal of Operational Research, {bf 62}textbf{(2)} (1992), 203-209.
92
bibitem{wang1997comparative}
93
X.~Wang, {it A comparative study of the ranking methods for fuzzy quantities}, Ghent University, Ghent, 1997.
94
bibitem{wang2001reasonable}
95
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), 375-385.
96
bibitem{wang2001reasonable2}
97
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), 387-405.
98
bibitem{yager1980general}
99
R. A. Yager, {it On a general class of fuzzy connectives}, Fuzzy sets and Systems, {bf 4}textbf{(3)} (1980), 235-242.
100
bibitem{yager1986paths}
101
R. A. Yager, {it Paths of least resistance in possibilistic production systems}, Fuzzy Sets and Systems, {bf 19}textbf{(2)} (1986), 121-132.
102
bibitem{zadeh1965fuzzy}
103
L. A. Zadeh, {it Fuzzy sets}, Information and control, {bf 8}textbf{(3)} (1965), 338-353.
104
ORIGINAL_ARTICLE
A FUZZY MINIMUM RISK MODEL FOR THE RAILWAY
TRANSPORTATION PLANNING PROBLEM
The railway transportation planning under the fuzzy environment
is investigated in this paper. As a main result, a new modeling method, called
minimum risk chance-constrained 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.
http://ijfs.usb.ac.ir/article_307_3d5c42e3f0fbf960235d85616888b3fc.pdf
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60
10.22111/ijfs.2011.307
Minimum risk model
Railway transportation planning
Credibility
measure
Discretization method
Lixing
Yang
lxyang@bjtu.edu.cn
true
1
State Key Laboratory of Rail Traffic Control and Safety, Beijing
Jiaotong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing
Jiaotong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing
Jiaotong University, Beijing 100044, China
LEAD_AUTHOR
Xiang
Li
xiang-li04@tsinghua.edu.cn
true
2
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
AUTHOR
Ziyou
Gao
gaoziyou@jtys.bjtu.edu.cn
true
3
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
AUTHOR
Keping
Li
likeping@jtys.bjtu.edu.cn
true
4
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China
AUTHOR
bibitem{Crainic2} T. G. Crainic and J. Roy, {it O. R. tools for tactical freight
1
transportation planning}, European Journal of Operational
2
Research, textbf{33(3)} (1988), 290-297.
3
bibitem{Gorman} M. F. Gorman, {it An application of genetic
4
and tabu searches to the freight railroad operating plan problem},
5
Annals of Operations Research, textbf{78} (1998), 51-69.
6
bibitem{Gorman2} M. F. Gorman, {it Santa fe railway uses an operating-plan model to improve its service
7
design}, Interfaces, textbf{28(4)} (1998), 1-12.
8
bibitem{Keaton} M. H. Keaton, {it Designing optimal railroad operating plans:
9
lagrangian relaxation and heuristic approaches}, Transportation
10
Research B, textbf{23(6)} (1989), 415-431.
11
bibitem{fuzzyexp}B. Liu and Y. K. Liu, {it Expected value of fuzzy variable and fuzzy expected value model}, IEEE
12
Transactions on Fuzzy Systems, textbf{10(4)} (2002), 445-450.
13
bibitem{liu2} B. Liu, {it Theory and practice of uncertain programming}, Physica-Verlag, Heidelberg, 2002.
14
bibitem{liu2004} B. Liu, {it Uncertainty theory: an introduction to its axiomatic foundations},
15
Springer-Verlag, Berlin, 2004.
16
bibitem{Liuyk1} Y. K. Liu, {it Convergence results about the use of fuzzy simulation in fuzzy optimization problems}, IEEE
17
Transactions on Fuzzy Systems, textbf{14} (2006), 295-304.
18
bibitem{Liuyk2} Y. K. Liu, {it The approximate method for two-stagefuzzy random programming with recourse}, IEEE
19
Transactions on Fuzzy Systems, textbf{15} (2007), 1197-1208.
20
bibitem{Liuyk3} Y. K. Liu, {it The convergence results about approximating fuzzy random minimum risk problems}, Applied
21
Mathematics and Computation, textbf{205(2)} (2008), 608-621.
22
bibitem{Nahmias} S. Nahmias, {it Fuzzy variable}, Fuzzy Sets and Systems,
23
textbf{1} (1978), 97-110.
24
bibitem{Newton} H. N. Newton, C. Barnhart and P. H. Vance, {it Constructing railroad
25
blocking plans to minimize handling costs}, Transportation
26
Science, textbf{32(4)} (1998), 330-345.
27
bibitem{qin2} Z. Qin and X. Gao, {it Fractional Liu process with application to finance},
28
Mathematical and Computer Modeling, textbf{50} (2009), 1538-1543.
29
bibitem{qin3} Z. Qin, X. Li and X. Ji, {it Portfolio selection based on fuzzy cross-entropy},
30
Journal of Computational and Applied Mathematics, textbf{228} (2009), 139-149.
31
bibitem{qin1} Z. Qin and X. Ji, {it Logistics network design for product recovery in fuzzy environment},
32
European Journal of Operational Research, textbf{202} (2010), 279-290.
33
bibitem{yang1} L. Yang and L. Liu, {it Fuzzy fixed charge solid transportation problem and algorithm},
34
Applied Soft Computing, textbf{7} (2007), 879-889.
35
bibitem{yang2} L. Yang, K. Li and Z. Gao, {it Train timetable problem on a single-line railway with
36
fuzzy passenger demand}, IEEE Transactions on Fuzzy Systems,
37
textbf{17(3)} (2009), 617-629.
38
bibitem{yang3} L. Yang, Z. Gao and K. Li, {it Railway freight transportation planning with mixed
39
uncertainty of randomness and fuzziness}, Applied Soft
40
Computing, textbf{11(1)} (2011), 778-792 .
41
bibitem{yang4} L. Yang, X. Ji, Z. Gao and K. Li, {it Logistics distribution centers
42
location problem and algorithm under fuzzy environment}, Journal
43
of Computational and Applied Mathematics, textbf{208(2)} (2007), 303-315.
44
bibitem{zadeh} L. A. Zadeh, {it Fuzzy sets}, Information and Control,
45
textbf{8} (1965), 338-353.
46
bibitem{zhou} J. Zhou and B. Liu, {it Convergence concept of bifuzzy sequence}, Asian Information-Science-Life,
47
textbf{2(3)} (2004), 297-310.
48
ORIGINAL_ARTICLE
MEAN-ABSOLUTE DEVIATION PORTFOLIO SELECTION
MODEL WITH FUZZY RETURNS
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 mean-absolute 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.
http://ijfs.usb.ac.ir/article_308_3b4ec0110e6e5007d107918ec346e5a6.pdf
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75
10.22111/ijfs.2011.308
Uncertainty modelling
Fuzzy variable
Fuzzy portfolio selection
Credibility theory
Hybrid intelligent algorithm
Zhongfeng
Qin
qin@buaa.edu.cn, qzf05@mails.thu.edu.cn
true
1
School of Economics and Management, Beihang University, Beijing
100191, China
School of Economics and Management, Beihang University, Beijing
100191, China
School of Economics and Management, Beihang University, Beijing
100191, China
AUTHOR
Meilin
Wen
wenmeilin@buaa.edu.cn
true
2
School of Reliability and Systems Engineering, Beihang University,
Beijing 100191, China
School of Reliability and Systems Engineering, Beihang University,
Beijing 100191, China
School of Reliability and Systems Engineering, Beihang University,
Beijing 100191, China
LEAD_AUTHOR
Changchao
Gu
guchangchao@gamil.com
true
3
Sinopec Management Institute, Beijing 100012, China
Sinopec Management Institute, Beijing 100012, China
Sinopec Management Institute, Beijing 100012, China
AUTHOR
bibitem{Abdelaziz} F. Abdelaziz, B. Aouni and R. Fayedh, {it
1
Multi-objective stochastic programming for portfolio selection},
2
Eur. J. Oper. Res., {bf 177} (2007), 1811-1823.
3
bibitem{Arenas-Parra} M. Arenas-Parra, A. Bilbao-Terol and M. Rodr'{i}guez-Ur'{i}a,
4
{it A fuzzy goal programming approach to portfolio selection}, Eur.
5
J. Oper. Res., {bf 133} (2001), 287-297.
6
bibitem{Bilbao} A. Bilbao-Terol, B. P'{e}rez-Gladish, M. Arenas-Parra and M. Rodr'{i}gez-Ur'{i}a,
7
{it Fuzzy compromise programming for portfolio selection}, Appl.
8
Math. Comput., {bf 173} (2006), 251-264.
9
bibitem{Corazza} M. Corazza and D. Favaretto, {it On the existence of
10
solutions to the quadratic mixed-integer mean-variance portfolio
11
selection problem}, Eur. J. Oper. Res., {bf 176} (2007), 1947-1980.
12
%bibitem{Deng} X. Deng, Z. Li and S. Wang, {it A minimax portfolio
13
%selection strategy with equilibrium}, Eur. J. Oper. Res., {bf
14
%166}(2005), 278--292.
15
%bibitem{Grootveld} H. Grootveld and W. Hallerbach, {it Variance vs
16
%downside risk: is there really that much difference?}, Eur. J. Oper.
17
%Res., {bf 114}(1999), 304--319.
18
%bibitem{huang chance constrained}X. Huang, {it Fuzzy chance-constrained
19
%portfolio selection}, Appl. Math. Comput., {bf 177}(2006),
20
%500--507.
21
%bibitem{Huang mean variance} X. Huang, {it Portfolio selection with
22
%fuzzy returns}, J. Intell. Fuzzy Syst., {bf 18}(2007), 383--390.
23
bibitem{huang mean semivariance} X. Huang, {it Mean-semivariance models
24
for fuzzy protfolio selection}, J. Comput. Appl. Math., {bf
25
217} (2008), 1-8.
26
bibitem{Konno and Yamazaki} H. Konno and H. Yamazaki, {it Mean-absolute
27
deviation portfolio optimization model and its applications to Tokyo
28
Stock Market}, Manage. Sci., {bf 37} (1991), 519-531.
29
bibitem{li and liu 2006} X. Li and B. Liu, {it A sufficient and necessary condition for credibility
30
measures}, Int. J. Uncertain. Fuzz., {bf 14} (2006), 527-535.
31
bibitem{Li and Liu entropy} X. Li and B. Liu, {it Maximum entropy
32
principle for fuzzy variables}, Int. J. Uncertain. Fuzz., {bf
33
15} (2007), 43-53.
34
bibitem{li and qin skewness} X. Li, Z. Qin and S. Kar, {it Mean-variance-skewness
35
model for portfolio selection with fuzzy returns}, Eur. J. Oper.
36
Res., {bf 202} (2010), 239-247.
37
bibitem{Liu book} B. Liu, {it Theory and practice of uncertain
38
programming}, Physica-Verlag, Heidelberg, 2002.
39
bibitem{liu 2007} B. Liu, {it Uncertainty theory}, 2nd ed.,
40
Springer-Verlag, Berlin, 2007.
41
bibitem{liu iwamura} B. Liu and K. Iwamura, {it Chance constrained
42
programming with fuzzy parameters}, Fuzzy Sets and Systems, {bf
43
94} (1998), 227-237.
44
bibitem{liu and liu 2002} B. Liu and Y. Liu, {it Expected value of fuzzy variable and
45
fuzzy expected value models}, IEEE T. Fuzzy Syst., {bf 10} (2002),
46
%bibitem{Liu wang} S. Liu, S. Wang and W. Qiu, {it A
47
%mean-variance-skewness model for portfolio selection with
48
%transaction costs}, Int. J. Syst. Sci, {bf 34}(2003), 255--262.
49
bibitem{ykliu1} Y. Liu, {it Convergent results about the use of fuzzy simulation in fuzzy optimization
50
problems}, IEEE T. Fuzzy Syst., {bf 14} (2006), 295-304.
51
bibitem{Markowitz 52} H. Markowitz, {it Porfolio selection}, J. Finance, {bf 7} (1952), 77-91.
52
bibitem{Markowitz 59} H. Markowitz, {it Portfolio selection: efficient
53
diversification of investments}, Wiley, New York, 1959.
54
%bibitem{Markowitz 90} H. Markowitz, {it Computation of
55
%mean-semivariance efficient sets by the critical line algorithm},
56
%Ann. Oper. Res., {bf 45}(1993), 307--317.
57
bibitem{Qin Li Ji} Z. Qin, X. Li and X. Ji, {it Portfolio selection based on
58
fuzzy cross-entropy}, J. Comput. Appl. Math., {bf 228} (2009),
59
%bibitem{Roy} A.D. Roy, {it Safety first and the holding of assets}, Econometrics, {bf 20}(1952), 431--449.
60
bibitem{Simaan} Y. Simaan, {it Estimation risk in portfolio selection:
61
the mean vairance model versus the mean absolute deviation model},
62
Manage. Sci., {bf 43} (1997), 1437-1446.
63
bibitem{Speranza} M. G. Speranza, {it Linear programming model for
64
portfolio optimization}, Finance, {bf 14} (1993), 107-123.
65
bibitem{Tanaka guo} H. Tanaka and P. Guo, {it Portfolio selection based on
66
upper and lower exponential possibility distributions}, Eur. J.
67
Oper. Res., {bf 114} (1999), 115-126.
68
bibitem{Tanaka guo Turksen} H. Tanaka, P. Guo and I. T"{u}rksen, {it Portfolio selection based on
69
fuzzy probabilities and possibility distributions}, Fuzzy Sets and
70
Systems, {bf 111} (2000), 387-397.
71
bibitem{Vercher} E. Vercher, J. Berm'{u}dez and J. Segura, {it Fuzzy
72
portfolio optimization under downside risk measures}, Fuzzy Sets and
73
Systems, {bf 158} (2007), 769-782.
74
bibitem{Yang 1} L. Yang, K. Li and Z. Gao, {it Train timetable
75
problem on a single-line railway with fuzzy passenger demand}, IEEE
76
T. Fuzzy Syst., {bf 17} (2009), 617-629.
77
bibitem{Yang 2} L. Yang and L. Liu, {it Fuzzy fixed charge solid transportation
78
problem and algorithm}, Appl. Soft Comput., {bf 7} (2007), 879-889.
79
bibitem{Yang 3} L. Yang, Z. Gao and K. Li, {it Railway
80
freight transportation planning with mixed uncertainty of randomness
81
and fuzziness}, Appl. Soft Comput., {bf 11} (2011), 778-792.
82
bibitem{Zadeh} L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf 8} (1965), 338-353.
83
bibitem{Zhang} W. Zhang and Z. Nie, {it On admissible efficient
84
portfolio selection problem}, Appl. Math. Comput., {bf 159} (2004),
85
ORIGINAL_ARTICLE
FUZZY TRAIN ENERGY CONSUMPTION MINIMIZATION
MODEL AND ALGORITHM
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 Karush-Kuhn-Tucker conditions. Finally, we illustrate
a numerical example to show the effectiveness of the proposed model and
algorithm.
http://ijfs.usb.ac.ir/article_309_b33ac36fdf8e3fc3c689b9438a056dfa.pdf
2011-10-07T11:23:20
2018-05-25T11:23:20
77
91
10.22111/ijfs.2011.309
Energy consumption
Train scheduling
Karush-Kuhn-Tucker condi-
tions
Fuzzy variable
Xiang
Li
xiang-li04@mail.tsinghua.edu.cn
true
1
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao-
tong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao-
tong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao-
tong University, Beijing 100044, China
LEAD_AUTHOR
Dan
Ralescu
ralescd@uc.edu
true
2
Department of Mathematical Sciences, University of Cincinnati, Cincin-
nati, Ohio 45221, USA
Department of Mathematical Sciences, University of Cincinnati, Cincin-
nati, Ohio 45221, USA
Department of Mathematical Sciences, University of Cincinnati, Cincin-
nati, Ohio 45221, USA
AUTHOR
Tao
Tang
ttang@bjtu.edu.cn
true
3
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao-
tong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao-
tong University, Beijing 100044, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao-
tong University, Beijing 100044, China
AUTHOR
[1] Y. Bai, B. Mao, F. Zhou, Y. Ding and C. Dong,Energy-ecient driving strategy for freight trains based on power consumption analysis, Journal of Transportation Systems Engineering and Information Technology,9(3) (2009), 43-50.
1
[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), 256-287.
2
[3] C. S. Chang and S. Sim,Optimising train movements through coast control using genetic algorithms, IEE Proceedings-Electric Power Applications, 144(1) (1997), 65-73.
3
[4] S. Eati and H. Roohparvar,The minimization of the fuel costs in the train transportation,Applied Mathematics and Computation,175 (2006), 1415-1431.
4
[5] K. Ghoseiri, F. Szidarovszky and M. J. Asgharpour,Amulti-objective train scheduling model and solution, Transportation Research Part B, 38 (2004), 927-952.
5
[6] P. Howlett,The optimal control of a train, Annals of Operations Research, 98 (2000), 65-87.
6
[7] P. Howlett, P. Pudney and X. Vu,Local energy minimization in optimal train control, Au-tomatica,45(11)(2009), 2692-2698.
7
[8] K. B. Khan and X. S. Zhou,Stochastic optimization model and solution algorithm for ro-bust double-track train-timetabling problem, IEEE Transactions on Intelligent Transportation Systems,11(1)(2010), 81-89.
8
[9] E. Khmelnitsky,On an optimal control problem of train operation, IEEE Transactions on Automatic Control,45(7) (2000), 1257-1266.
9
[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), 82-99.
10
[11] X. Li and B. Liu,A sucient and necessary condition for credibility measure, Internationa Journal of Uncertainty, Fuzziness & Knowledge-Based Systems,14(5) (2006), 527-535.
11
[12] P. Li and B. Liu,Entropy of credibility distributions for fuzzy variables, IEEE Transactions on Fuzzy Systems,16(1) (2008), 123-129.
12
[13] B. Liu and Y. Liu,Expected value of fuzzy variable and fuzzy expected value models, IEEETransactions on Fuzzy Systems,10(4) (2002), 445-450.
13
[14] R. Liu and I. M. Golovitcher,Energy-ecient operation of rail vehicles, Transportation Research Part A,37(2003), 917-932.
14
[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), 263-269.
15
[16] L. Yang, K. P. Li and Z. Y. Gao,Train timetable problem on a single-line railway with fuzzy passenger demand, IEEE Transactions on Fuzzy Systems, 17(3)(2009), 617-629.
16
[17] L. Yang, Z. Y. Gao and K. P. Li,Passenger train scheduling on a single-track or partially double-track railway with stochastic information, Engineering Optimization, 42(11) (2010),1003-1022.
17
ORIGINAL_ARTICLE
FLUENCE MAP OPTIMIZATION IN INTENSITY MODULATED
RADIATION THERAPY FOR FUZZY TARGET DOSE
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 non-fuzzy 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 head-and-neck cancers, each for two patients) are given.
http://ijfs.usb.ac.ir/article_310_624d131dd630762401d6a0a4997854fa.pdf
2011-10-07T11:23:20
2018-05-25T11:23:20
93
105
10.22111/ijfs.2011.310
IMRT
Singed distance
Triangular fuzzy number
Gradient method
Alireza
Fakharzadeh Jahromi
a_ fakharzadeh@sutech.ac.ir
true
1
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz, Fars, Iran
LEAD_AUTHOR
Omolbanin
Bozorg
o.bozorg@gmail.com
true
2
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz, Fars, Iran
AUTHOR
Hamidreza
Maleki
maleki@sutech.ac.ir
true
3
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz, Fars, Iran
Shiraz University of Technology, Shiraz, Fars, Iran
AUTHOR
Mohamad Amin
Mosleh-Shirazi
mosleh_amin@hotmail.com
true
4
Shiraz University of Medical Sciences, Shiraz, Fars,
Shiraz University of Medical Sciences, Shiraz, Fars,
Shiraz University of Medical Sciences, Shiraz, Fars,
AUTHOR
bibitem{2} G. Bahr, J. Kereiakes, H. Horwitz, R. Finnery, J. Calvin and
1
K. Goode, {it The method of linear programming applied to
2
radiation treatment planning}, Radiology, {bf 91} (1968), 686-693.
3
bibitem{salmani ref1} J. Chiang, {it Fuzzy linear programming based on statistical confidence interval and
4
interval-valued fuzzy set}, European Journal of
5
Operational Research, {bf 129} (2001), 65-86.
6
bibitem{8} C. Cotrutz, M. Lahanas, C. Kappas and D. Baltas, {it A
7
multiobjective gradient based dose optimization algorithm for
8
external beam conformal radiotherapy}, Physics in Medicine and Biology, {bf 46} (2001), 2161-2175.
9
bibitem{CERR} J. Deasy, A. Blanco and V. Clark, {it Cerr: a computational environment for radiotherapy research}, Medical Physics, {bf 30} (2003), 979-985.
10
bibitem{Dubois} D. Dubois and H. Prade, {it Fuzzy sets and systems: theory and application},
11
Academic Press, New York, 1980.
12
bibitem{Hamacher} H. W. Hamacher and K. H. Kufer, {it Inverse radiation therapy planninng - a multiple objective optimization
13
approach}, Discrete Applied Mathematics, {bf 118} (2002), 145-161.
14
bibitem{15} C. L. Lawson and R. J. Hanson, {it Solving least squares problems}, SIAM, (1995), 160-165.
15
bibitem{mixed integer} E. Lee, T. Fox and I. Crocker, {it Integer programming applied to intensity
16
modulated radiation therapy treatment planning}, Annals of
17
Operations Research, {bf 119} (2003), 165-181.
18
bibitem{IMRT1} M. S. Merritt, {it A sensitivity-driven greedy approach to fluence map
19
optimization in intensity-modulated radiation therapy}, Ph. D. thesis, Department of Computational and Applied Mathematics,
20
Rice University, April 2006.
21
bibitem{26} S. Morrill, R. Lane, J. Wong and I. Rosen, {it Dose-volume
22
considerations with linear programming optimization}, Medical Physics, {bf 18} (1991), 1201-1210.
23
bibitem{Preciado} F. Preciado-Walters, M. Langer, R. Rardin and V. Thai, {it A coupled column generation, mixed-integer
24
approach to optimal planning of intensity modulated therapy for
25
cancer}, Mathematical Programming, {bf 101} (2004), 319-338.
26
bibitem{Romeijn} H. E. Romeijn, R. K. Ahuja, J. F. Dempsey, A. Kumar and G. J. Li, {it Novel linear programming approach
27
to fluence map optimization for intensity modulated radiation
28
therapy treatment planning}, Physics in Medicine and Biology, {bf 48} (2003), 3521-3542.
29
bibitem{30} H. E. Romeijn, J. F. Dempsey and J. G. Li, {it A unifying framework for multi
30
criteria fluence map optimization models}, Physics in Medicine and Biology, {bf 49} (2004), 1991-2013.
31
bibitem{34} S. V. Spirou and C. Chui, {it A gradient inverse planning algorithm with dose-volume constraints}, Medical Physics, {bf 25} (1998), 321-333.
32
bibitem{Yao wu} J. Yao and K. Wu, {it Ranking fuzzy numbers based on decomposition principle and signed
33
distance}, Fuzzy Sets and Systems, {bf 116} (2000), 275-288.
34
bibitem{Yin Zhang} Y. Zhang and M. Merrit, {it Dose-volume-based IMRT fluence optimization: a fast
35
leastsquares approach with differentiability}, Linear Algebra and
36
its Applications, {bf 428} (2008), 1365-1387.
37
bibitem{zimerman} H. J. Zimmermann, {it Fuzzy set theory and its applications},
38
second ed., Kluwer Academic Publishers, Boston, 1991.
39
ORIGINAL_ARTICLE
COALITIONAL GAME WITH FUZZY PAYOFFS AND
CREDIBILISTIC SHAPLEY VALUE
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.
http://ijfs.usb.ac.ir/article_311_ec8eadf2c503f1494486ac8555956b9a.pdf
2011-10-07T11:23:20
2018-05-25T11:23:20
107
117
10.22111/ijfs.2011.311
Coalitional game
Shapley value
Fuzzy variable
Credibility measure
Jinwu
Gao
jgao@ruc.edu.cn
true
1
Uncertain Systems Lab, School of Information, Renmin University of
China, Beijing 100872, China
Uncertain Systems Lab, School of Information, Renmin University of
China, Beijing 100872, China
Uncertain Systems Lab, School of Information, Renmin University of
China, Beijing 100872, China
LEAD_AUTHOR
Q.
Zhang
zqw2002@163.com
true
2
Uncertain Systems Lab, School of Information, Renmin University of
China, Beijing 100872, China
Uncertain Systems Lab, School of Information, Renmin University of
China, Beijing 100872, China
Uncertain Systems Lab, School of Information, Renmin University of
China, Beijing 100872, China
AUTHOR
P.
Shen
shenpuchen@163.com
true
3
Uncertain Systems Lab, School of Information, Renmin University of China,
Beijing 100872, China
Uncertain Systems Lab, School of Information, Renmin University of China,
Beijing 100872, China
Uncertain Systems Lab, School of Information, Renmin University of China,
Beijing 100872, China
AUTHOR
bibitem{Aumann64} R. Aumann and M. Maschler, {it The bargainning set
1
for cooperative games}, Advances in Game Theory, Annals of
2
Mathematics Studies, {bf 52} (1964), 443-476.
3
bibitem{Aumann74} R. Aumann and L. S. Shapley, {it Values of non-atomic games}, Princeton University,
4
Princeton, 1974.
5
bibitem{Aubin81} J. P. Aubin, {it Cooperative fuzzy games}, Mathematical Operation
6
Research, {bf 6} (1981), 1-13.
7
bibitem{Butnariu78} D. Butnariu, {it Fuzzy games: a description of the
8
concept}, Fuzzy Sets and Systems, {bf 1} (1978), 181-192.
9
bibitem{Butnariu80} D. Butnariu, {it Stability and shapley value for an n-persons fuzzy games}, Fuzzy Sets and Systems, {bf 4} (1980), 63-72.
10
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), 149-160.
11
bibitem{Campos92} L. Campos, A. Gonzalez and M. A. Vila, {it On the use of the ranking function approach to solve fuzzy
12
matrix games in a direct way}, Fuzzy Sets and Systems, {bf 49} (1992), 193-203.
13
bibitem{dubios88} D.~Dubois and H.~Prade, {it Possibility theory},
14
Plenum, New York, 1988.
15
bibitem{Gao05} J.~Gao, and B.~Liu, {it Fuzzy multilevel programming with a hybrid intelligent
16
algorithm}, Computer & Mathmatics with applications, {bf 49} (2005), 1539-1548.
17
bibitem{Gao07} J. Gao, {it Credibilistic game with fuzzy
18
information}, Journal of Uncertain Systems, {bf 1(1)} (2007), 72-78.
19
bibitem{Gao09} J. Gao, Z. Q. Liu and P. Shen, {it On characterization of fuzzy-payoff two-player zero-sum game}, Soft Computing, {bf 13(2)} (2009), 127-132.
20
bibitem{Gao111} J. Gao, {it Fuzzy dependent-chance multilevel programming with applications}, Journal of Universal Computer Science, to be Published.
21
bibitem{Gao112} J. Gao and Y. Yu, {it Credibilistic extensive game with fuzzy information}, Soft Computing, to be Published.
22
bibitem{Harsanyi95} J. C. Harsanyi, {it Games with incomplete information},
23
The American Economic Review, {bf 85(3)} (1995), 291-303.
24
bibitem{Gao101} R. Liang, Y. Yu, J. Gao and Z. Q. Liu, {it N-person credibilistic strategic game}, Frontiers of Computer Science in China, {bf 4(2)} (2010), 212-219.
25
bibitem{Liu02} B.~Liu and Y.~Liu, {it Expected value of fuzzy variable and fuzzy
26
expected value models}, IEEE Transactions on Fuzzy Systems, {bf 10} (2002), 445-450.
27
bibitem{Liu03} Y. Liu and B. Liu, {it Expected value operator of random fuzzy variable and random
28
fuzzy expected value models}, International Journal of
29
Uncertainty, Fuzziness & Knowledge-Based Systems, {bf 11(2)} (2003), 195-215.
30
bibitem{Liubook04} B.~Liu, {it Uncertainty theory: an introduction to its axiomatic
31
foundations}, Springer-Verlag, Berlin, 2004.
32
bibitem{Liu06} B. Liu, {it A survey of credibility theory}, Fuzzy Optimization and Decision Making,
33
{bf 5(4)} (2006), 387-408.
34
bibitem{Liu07} Y. Liu and J. Gao, {it The dependence of fuzzy variables with applications
35
to fuzzy random optimization}, International Journal of
36
Uncertainty, Fuzziness & Knowledge-Based Systems, {bf 15} (2007), 1-20.
37
bibitem{Liubook07} B.~Liu, {it Uncertainty theory}, 2nd Edition, Springer-Verlag, Berlin, 2007.
38
bibitem{Neumann44} J. Von Neumann and D. Morgenstern, {it The theory of games in economic bahavior},
39
New York: Wiley, 1944.
40
bibitem{Maeda00} T. Maeda, {it Characterization of the equilibrium strategy of the bimatrix
41
game with fuzzy payoff}, Journal of Mathematical Analysis and
42
Applications, {bf 251} (2000), 885-896.
43
bibitem{Maeda03} T. Maeda, {it Characterization of the equilibrium strategy of the two-person
44
zero-sum game with fuzzy payoff}, Fuzzy Sets and Systems, {bf 139} (2003), 283-296.
45
bibitem{Mares94} M. Mareu{s}, {it Computation over fuzzy quantities}, CRC-Press, 1994.
46
bibitem{Mares00} M. Mareu{s}, {it Fuzzy coalitions structures}, Fuzzy Sets
47
and Systems, {bf 114(1)} (2000), 23-33.
48
bibitem{Mares01} M. Mareu{s}, {it Fuzzy cooperative games}, Physica-Verlag, Heidelberg, 2001.
49
bibitem{Maschler79} M. Mashler, B. Peleg and L. S. Shapley, {it Geometric
50
properties of the kernel, nucleolus and related solution concepts},
51
Mathematics Operational Research, {bf 4} (1979), 303-337.
52
bibitem{Nishizaki00} I. Nishizaki and M. Sakawa, {it Equilibrium solutions for multiobjective bimatrix games with fuzzy payoffs
53
and fuzzy goals}, Fuzzy Sets and Systems, {bf 111(1)} (2000), 99-116.
54
bibitem{Nishizaki00a} I. Nishiizaki and M. Sakawa, {it Fuzzy cooperative games
55
arising from linear production programming problems with fuzzy
56
parameters}, Fuzzy sets and Systems, {bf 114(1)} (2000), 11-21.
57
bibitem{Nishizaki00b} I. Nishiizaki and M. Sakawa, {it Solutions based
58
on fuzzy goals in fuzzy linear programming games}, Fuzzy Sets
59
and Systems, {bf 115(1)} (2000), 105-109.
60
bibitem{Shapley53} L. S. Shapley, {it A value for n-persons games}, Annals of Mathematics Studies, {bf 28} (1953), 307-318.
61
bibitem{Shapley54} L. S. Shapley and M. Shubik, {it A method for evaluating
62
the distribution of power in committee systerm}, American
63
Politics Science Review, {bf 48} (1954), 787-792.
64
bibitem{Schmeidler69} D. Schmeidler, {it The nucleolus of a
65
characteristic function games}, Journal of Applied Mathematics
66
{bf 17} (1969), 1163-1170.
67
bibitem{Gao102} P. Shen and J. Gao, {it Coalitional game with fuzzy information and credibilistic core}, Soft Computing, DOI: 10.1007/s00500-010-0632-9, 2010.
68
%bibitem{Peleg03} B. Peleg, {em Introdutin to the theory of cooperative
69
%game}, Kluswer Academic Publishers, Boston, 2003.
70
bibitem{zadeh65} L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf 8}(1965), 338-353.
71
bibitem{zadeh78} L. A. Zadeh, {it Fuzzy sets as a basis for a theory of
72
possibility}, Fuzzy Sets and Systems, {bf 1}(1978), 3-28.vspace{-.3 cm}
73
ORIGINAL_ARTICLE
Persian-translation vol. 8, no. 4, October 2011
http://ijfs.usb.ac.ir/article_2869_0ffa9f8030e868147b8d43909fd9e0de.pdf
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2018-05-25T11:23:20
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10.22111/ijfs.2011.2869