This paper proposes a mathematical programming approach for membership and non-member functions of steady-state performance measures in infinite capacity queues in the Erlang service, where the arrival rate and service rate are intuitionistic fuzzy numbers. The rudimentary idea is based on an Atanassov’s extension principle and (α, β)- cut approach. Two pairs of mixed-integer nonlinear programs(MINLP) with binary variables are regulated to calculate the upper and lower bounds of the system performance measure of possibility level (α, β). The main objective of the paper is to investigate the expected and awaiting number of customers in the queue and their waiting time in service. Finally, a numerical example is illustrated to show the efficiency of the proposed method.
Topics
Fuzzy numbers
REFERENCES
1.
V. Ashok
Kumar
, “A membership function solution approach to fuzzy queue with Erlang service model
”, Internatinal Journal of Mathematical Science and Applications
1
,2
, 881
–891
(2011
).2.
K.T.
Atanassov
, Intuitionistic Fuzzy Sets, (VII ITKR’s Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84, in Bulgarian
) 1983
).3.
K.T.
Atanassov
, “Intuitionistic Fuzzy Sets
”, Fuzzy Sets and Systems
, 20
, 87
–96
(1986
).4.
K.T.
Atanassov
, “More on Intutionistic fuzzy sets
”, Fuzzy Sets and Systems 33
, 1
, 37
–46
(1981
).5.
K.T.
Atanassov
, Intuitionistic Fuzzy Sets
, (Heidelberg, New York
: Physica-Verlag, Berlin
1999
).6.
A.
Borthakur
, “A Poisson queue with general bulk service rule
”, Journal of the Assam Scientific Society
14
, 162
–167
(1971
).7.
J.J.
Buckley
, “Elementary Queuing Theory based on Possibility Theory
”, Fuzzy Sets and System
37
, 43
–52
(1990
) .8.
J.J.
Buckley
, T.
Feuring
, Y.
Hasaki
, “Fuzzy Queueing Theory Resisted
”, International Journal of Uncertainty Fuzzy
9
, 527
–537
(2001
).9.
Chaiang
Kai
, Chang-Chung
Li
, Shin-Pin
Chin
, “Parametric programming to analysis of fuzzy queues
”, Fuzzy Sets and Systems
107
, 93
–100
(1999
).10.
S.
Chanas
, “Parametric programming in fuzzy linear programming
”, Fuzzy Sets and Systems
11
, 243
–251
(1983
) .11.
S.
Chanas
, M.
Nowakowski
, “Single value simulation of fuzzy variable
”, Fuzzy Sets and Systems
21
, 43
–57
(1988
).12.
S.P.
Chen
, “Parametric nonlinear programming for analyzing fuzzy queues with finite capacity
”, European Journal of Operational Research
157
, 429
–438
(2004
).13.
S.P.
Chen
, “Parametric nonlinear programming approach to fuzzy queues with bulk service
”, European Journal of Operational Research
163
, 434
–444
(2005
).14.
S.P.
Chen
, “A bulk arrival queueing model with fuzzy parameters and varying batch sizes
”, Applied Mathematical Modelling
30
, 920
–928
(2006
).15.
16.
G.
Deschrijver
, E.E
Kerre
, “On the relationship between intuitionistic fuzzy sets and some other extensions of fuzzy set theory
”, Journal of Fuzzy Mathematics
10
, 3
, 711
–724
(2001
).17.
D.
Gross
, C.M.
Haris
, Fundamentals of Queueing Theory
, (Wiley
, New York
, 1998
, 3rd Ed).18.
J.B.
Jo
, Y.
Tsujimura
, M.
Gon
, G.
Yamazaki
, “Performance Evaluation of Network Models based on Fuzzy Queueing System
”, Japan Journal of Fuzzy Theory System
8
, 393
–408
(1996
).19.
A.
Kaufmann
, Introduction to the Theory of Fuzzy subsets
, (Academic Press
, New York
, 1974
).20.
R.J.
Li
, E.S.
Lee
, “Analysis of fuzzy queue
”, Computers and Mathematics with Applications
, 17
, 1143
–1147
(1984
).21.
D.S.
Nagi
, E.S.
Lee
, “Analysis and simulation of fuzzy queue
”, Fuzzy Sets and Systems
46
, 321
–330
(1992
).22.
K.
Punniakrishnan
, K.
Kadambavanam
, “On Intuitionistic Fuzzy Inventory Models Without Allowing storage Constraint
”, Impact: International Journal of Research in Engineering and Technology
2
, 155
–166
(2014
).23.
H.M.
Prade
, An outline of fuzzy or possibilistic models for queueing system, in : P.P.
Wong
, S.K.
Chang
(Eds).,(Fuzzy Sets Plenum Press
, New York
, 147
–153
, 1980
).24.
P.
Rajarajeswari
, M.
Sangeetha
, “Fuzzy intuitionistic on queueing system
”, International journal of Fuzzy Mathematics and Systems
4
, 105
–119
(2014
).25.
W.
Ritha
, B. Sreelekha
Menon
, “Analysis of fuzzy Erlang’s loss queueing model:non linear programming approach
”, International Journal of Fuzzy Mathematics and Systems
1
, 1
–10
(2011
).26.
Sankar Prasad
Mondal
, Tapan Kumar
Roy
, Non-linear arithmetic operation on generalized triangular intuitionistic fuzzy numbers
, Notes on intuitionistic fuzzy sets
20
, pp.9
–19
(2014
).27.
A.K.
Show
, T.K.
Roy
, “Trapezoidal intuitionistic fuzzy number with some arithmetic operations and its application on reliability evaluation
”, International Journal of Mathematics in Operations Research
, 5
, 55
–73
(2013
).28.
R.
Srinivasan
, “Fuzzy Queueing Model Using DSW algorithm
”, International Journal of Advanced research in Mathematics and Applications1
, 1
, 57
–62
(2014
).29.
R.R.
Yager
, “A characterization of the extension principle
”, Fuzzy Sets and Systems
18
, 205
–217
(1986
).30.
L.A.
Zadeh
, “Fuzzy Sets as a Basis for a Theory of Possibility
”, Fuzzy Sets and Systems
1
, 3
–28
(1978
).31.
H.J.
Zimmermann
, Fuzzy Set Theory and its Applications, (Kluwar Academic
, Boston
, 2001
), 4th ed.
This content is only available via PDF.
© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
