4.1.1 First Layer
In this section, all input parameters and their mathematical relationship are defined.
Fig. 6 shows the vehicle lane variable consisting of three membership functions. The line
with a circle marker represents the vehicle in the road's right lane. The line with
the "x" marker represents the center lane, and the line with the diamond marker represents
the left lane. The association of vehicles with the lane is represented by the degree
of membership. The degree of the membership function is always defined from 0 to 1.
The mathematical form of the vehicle lane variable is given below.
Fig. 6. Membership Function for Vehicle Lane.
In Eq. (1), $\mu _{vl,R}\left(vl\right)$ represents the vehicle in the right lane.
In Eq. (2), $\mu _{vl,C}\left(vl\right)~ $represents the vehicle lane membership function with
the center lane.
In Eq. (3), $\mu _{vl,L}\left(vl\right)$ represents the vehicle membership function with the
left lane.
Fig. 7 shows the vehicle type variable used to distinguish between public transport, private
transport, and emergency vehicles. The vehicle type variable is comprised of three
membership functions. Public transport is represented using the "o" marker, private
transport is expressed with the "x" marker, and emergency vehicles are represented
by the "${\lozenge}$" marker.
In Eq. (4), $\mu _{vt,Publictrans}\left(vt\right)$represents the membership function for the
public transport vehicle.
In Eq. (5), $\mu _{vt,pritrans}\left(vt\right)$ represents the membership function for the private
transport vehicle.
Eq. (6) shows the membership function of an emergency vehicle. $\mu _{vt,emergency}\left(vt\right)~
$represents the emergency vehicle.
Fig. 7. Vehicle Type Membership Function.
Fig. 8 shows the ``distance from the IMS'' variable. The line with the "o" marker represents
a low distance ranging from 0 to 5 meters, and the line with the "x" marker represents
a medium distance ranging from 4 to 10 meters. The line with the "${\lozenge}$" marker
represents a vehicle that is a little far from the IMS.
Fig. 8. Distance Membership Function.
In Eq. (7), $\mu _{vd,~ low}\left(vd\right)$ represents the low distance from the IMS.
In Eq. (8) $\mu _{vd,medium}\left(vd\right)$ represents the medium-distance membership function
from the IMS.
In Eq. (9), $\mu _{vd,high}\left(vd\right)$ represents the high distance from the IMS membership
function.
Fig. 9 shows the graph of the estimated time for a vehicle to reach the IMS, which is divided
into three categories. The units used in this membership function are seconds. The
line with the "o" marker shows a low-time membership function and ranges from 0-10
seconds. The line with the "x" marker shows the medium time required to reach the
IMS, ranging from 8-20 seconds. The line with the "${\lozenge}$" marker shows the
high time needed to go to the IMS, ranging from 18-30 seconds. The mathematical form
of the membership function is given below.
Fig. 9. Estimate time to reach IMS.
In Eq. (10), $\mu _{est,low}\left(et\right)$ shows the low-estimate-time membership function.
In Eq. (11), $\mu _{est,medium}\left(est\right)~ $represents the membership function for the
medium estimate time from the IMS.
In Eq. (12), $\mu _{est,high}\left(est\right)$ represents the membership function for the high
estimate time to reach the IMS.
Fig. 10 represents the IMS's output variable at the first layer. The output variable consists
of three membership functions. The line with the "o" marker represents the leave membership
function. The line with marker "x" represents the standby action by the IMS for the
vehicle. The line with the "${\lozenge}$" marker represents the stop membership function.
The mathematical form of the IMS action membership function is given below. In Eq.
(13), $\mu _{IMS-action,leave}\left(lve\right)$ represents the leave membership function
of the IMS action variable.
In Eq. (14), $\mu _{IMS-action}\left(sb\right)$ represents the standby membership function of
the IMS action variable.
In Eq. (15), $\mu _{IMS-action}\left(st\right)$ represents the stop membership function of the
IMS action variable.
4.1.2 Second Layer
All the IMS output (i.e., IMS1, IMS2, IMS3, and IMS4 output) is fed to the CIMS as
input. A graph of individual IMSs is shown in Fig. 10. The mathematical form of all membership functions of the IMSs is given in Eqs. (13) and (14). The CIMS action variable contains four membership functions: Leave-IMS1, Leave-IMS2,
Leave-IMS3, and Leave-IMS4. The CIMS finally decides which vehicle should leave the
intersection. Fig. 11 shows the graph of output from the CIMS.
Fig. 10. IMS output at the first layer.
These membership functions are expressed mathematically in Eq. (16).
$\mu _{CIMS,leave-IMS1}\left(lm\right)$ represents the leave-IMS1 membership function.
$\mu _{CIMS,leave-IMS2}\left(lm\right)$ represents the leave-IMS2 membership function.
$\mu _{CIMS,leave-IMS3}\left(lm\right)$ represents the leave-IMS3 membership function.
$\mu _{CIMS,leave-IMS4}\left(lm\right)$ represents the leave-IMS4 membership function.