Transactions on Industrial Informatics

part have to be taken account.
𝜕𝐹
𝜕𝑝
𝑣
=
∑
𝜕𝐹
𝜕𝑓
𝑖
𝜕𝑓
𝑖
𝜕𝑝
𝑣
𝑣 ∈𝐶𝑜𝑛(𝑅
𝑖)
=
1
∑
𝑤
𝑗
20
𝑗=1
∑
𝑤
𝑖
𝜕𝑓
𝑖
𝜕𝑝
𝑣
𝑣 ∈𝐶𝑜𝑛(𝑅
𝑖)
(11)
In Eq. (11), 
∑
𝑤
𝑗
20
𝑗=1
is the summation of firing strength of all 
the rules in rule base and ∂f
i
/∂p
v
is based on the form of 
membership function and the defuzzification method. 
For antecedent labels, the calculation is similar but it requires 
a few more pass of the derivative of chain rule. The action 
depends on the degrees 
w
i
, which in turn depend on the 
membership degree 
μ
i
generated in layer 2. 
𝜕𝐹
𝜕𝑝
𝑣
=
𝜕𝐹
𝜕𝜇
𝑣
𝜕𝜇
𝑣
𝜕𝑝
𝑣
= (
∑
𝜕𝐹
𝜕𝑤
𝑖
𝜕𝑤
𝑖
𝜕𝜇
𝑣
𝑣 ∈𝐴𝑛𝑡(𝑅
𝑖)
)
𝜕𝜇
𝑣
𝜕𝑝
𝑣
(12)
𝜕𝐹
𝜕𝑤
𝑖
=
𝑓
𝑖
+ 𝑤
𝑖
𝜕𝑓
𝑖
𝜕𝑤
𝑖
− 𝐹
∑
𝑤
𝑗
20
𝑗=1
(13)
Similarly, 
∑
𝑤
𝑗
20
𝑗=1
is the sum of firing strength of all the rules 
in rule base. ∂
f
i
/∂
w
i
depends on defuzzification method and 
∂
μ
v
/∂
p
v
is based on the form of membership function. Since we 
use 
softmin
operation to get firing strength, ∂
w
i
/∂
μ
v
can be 
computed as follow: 
𝜕𝑤
𝑖
𝜕𝜇
𝑣
=
𝑒
−𝑘𝜇
𝑣
(1 + 𝑘(𝑤
𝑖
− 𝜇
𝑣
))
∑ 𝑒
−𝑘𝜇
𝑖
𝑖
(14)
where, 
∑ 𝑒
−𝑘𝜇
𝑖
𝑖
is calculated by going through 
membership degrees of all the antecedent linguistic labels in 
rule 
i
. 
D.
 
Group Scheduling 
The group scheduling part of our algorithm is used to grant 
permit of passing intersection to vehicles. The pseudo code of 
the algorithm are listed as Algorithm 1. Obviously, only the 
head groups at different lanes are candidates for the next 
passing. The metric of urgency degree is used to select next 
group.
The algorithm first computes urgency degree of each head 
group using fuzzy logic and the head group with the highest 
urgency degree is selected as the next to pass. The urgency 
degrees of two concurrent groups are then compared and the 
one with higher degree is also granted with permit. 
Two fuzzy variables are used as input in group scheduling, 
i.e. group size (GSZ) and average waiting time (AWT). The 
output is urgency degree (UD). The fuzzy rule base is shown in 
Table II. The structure and learning algorithm for the neuro-
fuzzy is similar to that of vehicle grouping. 
After the next group and its concurrent group are selected, 
the controller will broadcast a PERMIT message to vehicles. 
This message contains the list of two groups of vehicles being 
granted permit, denoted by plt.