Let [M.sub.1] = ([l.sub.1], [m.sub.1] - [u.sub.1]) and [M.sub.2] = ([l.sub.2], [m.sub.2], [u.sub.2]) be two TFNs
, whereby the degree of possibility of [M.sub.1] [greater than or equal to] [M.sub.2] is defined as follows:
According to fuzzy extent analysis, the method can be performed with respect to each object for each corresponding goal, [g.sub.i], resulting in m extent analysis values for each object, given as [M.sup.1.sub.gi], [M.sup.2.sub.gi], ..., [M.sup.n.sub.gi], i = 1,2, ...,n where all the [M.sup.j.sub.gi](j = 1,2, ...,m) are TFNs
representing the performance of the object [x.sub.i] with regard to each goal [u.sub.j].
Decision makers use predefined linguistic expressions which are modelled by TFNs
. The aggregated values of criteria values for suppliers over time period are given by using fuzzy averaging method.
There are miscellaneous types of fuzzy membership functions that triangular fuzzy number (TFN
) is one of them (see Fig.
Lower limit matrix (*) Most probable matrix (*) Factors FR B II CI MS FR B II CI MS FR 1 41/2 41/2 2/15 2/15 1 5 5 1/7 1/7 B 2/11 1 1 1/9 1/9 1/5 1 1 1/9 1/9 II 2/11 1 1 1/9 1/9 1/5 1 1 1/9 1/9 CI 61/2 81/2 81/2 1 1 7 9 9 1 1 MS 61/2 81/2 1/9 1 1 7 9 1/9 1 1 Lower limit matrix (*) Factors FR B II CI CI FR 1 51/2 51/2 2/13 2/13 B 2/9 1 1 2/17 2/17 II 2/9 1 1 2/17 2/17 CI 71/2 9 9 1 1 MS 71/2 9 2/17 1 1 (*) Lower, Most probable and upper limit matrices values are as per the adjusted Triangular Fuzzy Number (TFN
) matrix:; [??] Table 5.
In general, two types of fuzzy numbers, triangular and trapezoidal, are used, in which triangular fuzzy number (TFN
) is commonly used for computation efficient information in fuzzy environment.
Considering the uncertainty and vagueness of practical applications, a more appropriate way of representing values of indicators with Linguistic Variables which can further transform into Triangular Fuzzy Numbers (TFNs
) is presented.
In this paper, we used the TFNs
for its simplicity of computation, comparing to other fuzzy numbers, as presented in Equation (49) (Alavidoost et al., 2015b; Alavidoost et al., 2014; Alavidoost, 2017):
The elements of fuzzy pairwise comparison matrix should be described by linguistic expressions which are modelled by triangular fuzzy numbers (TFNs
A triangular fuzzy number (TFN
) [??] is exposed in "Eq.
The basic algebraic operations of addition, subtraction, multiplication and division on two TFNs
represented as [[??].sub.1] = ([a.sub.1], [b.sub.1], [c.sub.1]) and [[??].sub.2] = ([a.sub.2], [b.sub.2], [c.sub.2]) can be expressed as follows:
The report also outlines TfNs
implementation plan for Smart North, a smart ticketing system to simplify fares, slash queuing times and help passengers switch easily between buses, trains and trams.