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AdvancesinAppliedMathematicsA^êÆ?Ð,2019,8(7),1212-1223
PublishedOnlineJuly2019inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2019.87140
ANewAlgorithmforTrafficEquilibrium
FlowwithCapacityConstraintsofArc
DaqiongZhou,ZhiLin
∗
,ZaiyunPeng,JingjingWang
CollegeofMathematicsandStatistics,ChongqingJiaotongUniversity,Chongqing
Received:June30
th
,2019;accepted:July15
th
,2019;published:July22
nd
,2019
Abstract
Inthispaper,wemainlyresearchthealgorithmoftrafficequilibriumflowwithcapacity
constraints of arcs,and obtainthenecessary andsufficient condition thatfeasibleflow
xisatrafficequilibriumflowwithcapacityconstraintsofarcsbythedefinitionof
thedropoffeasibleflowx,anewalgorithmoftrafficequilibriumflowwithcapacity
constraintsofarcsisconstructed,andtheconcretestepsofcalculatingthetraffic
equilibriumflowwithcapacityconstraintsofarcsaregiven,atthesametime,an
exampleisgiventoillustratetheNewAlgorithm.
Keywords
Drop,NewAlgorithm,ArcCapacity,SaturationPath,TrafficEquilibriumProblem
withCapacityConstraintsofArc
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)
2
+45§
t
e13
(x
e13
)=3(x
e13
)
2
+15§t
e14
(x
e14
)=5(x
e14
)
2
+22§t
e15
(x
e15
)=(x
e15
)
2
+48§t
e16
(x
e16
)=2(x
e16
)
2
+55§
t
e17
(x
e17
)=2(x
e17
)
2
+66§t
e18
(x
e18
)=3(x
e18
)
2
+13§t
e19
(x
e19
)=3(x
e19
)
2
+25§t
e20
(x
e20
)=3(x
e20
)
2
+60§
t
e21
(x
e21
)=4(x
e21
)
2
+38§t
e22
(x
e22
)=2(x
e22
)
2
+70§t
e23
(x
e23
)=3(x
e23
)
2
+90"
e¡^#Ž{OŽälNþåÏþï6µ
i=0§
1)§éO/D:éω1=(1,12)§?À´»l
1
={e
3
e
10
e
18
e
23
}§-Ù6þ•x
1
(0)=6.00§éO/D:
éω2=(3,10)§?À´»l
2
={e
7
e
14
e
20
e
22
}§-Ù6þ•x
2
(0)=5.00"w,x(0)3l
1
þ6þx
1
(0) =
6.00§3l
2
þ6þx
2
(0)=5.00§Ù§´»þ6þ•0.00§H(0)={l
1
,l
2
}§Ù¥x
1
(0),x
2
(0)•
´»l
1
§l
2
6þ"dž3x(0)eä6þã„ã2()ÒSêi•lþ6þ)"
±x(0)•6ˆlþ|G•§\ä
b
ℵ
0
§dž\äã„ã3()ÒSêi•Œ1
6x(0)ž§ˆlþžm)"
2)§(1)d\äãã3§éO/D:éω1=(1,12)§ω
1
∈W§T
ω 1
(0)=t
l1
(x(0))=656.00¶
éO/D:éω2=(3,10)§ω
2
∈W§T
ω2
(0) = t
l2
(x(0)) = 482.00"
DOI:10.12677/aam.2019.871401217A^êÆ?Ð
±Œ 
Figure2.Thefigureofnetworkflowunderx(0)
ã2.3x(0)eä6þã
(2)d\äãã3§éO/D:éω1=(1,12)§^lingo^‡(±eOŽ•á´9gá´þ
æ^lingo^‡)ŽÑ•á´S
ω
1
={l
3
}={e
1
e
6
e
14
e
21
}§
e
T
ω1
(0)=t
l3
(x(0))=236.00¶éO/D:éω2
=(3,10)§•á´S
ω
2
= {l
4
}={e
2
e
5
e
11
e
17
}§
e
T
ω2
(0)=t
l4
(x(0))=161.00"
Ω
ω1
(0) = max{0,656−236}= 480.00§Ω
ω2
(0) = max{0,482−161}= 321.00"
Ω(0) = max{480.00,321.00}= 480.00"
3¤§ω1∗∈{ω∈W: Ω
ω1
(0) = Ω(0)}§á:é•ω1∗= (1,12)"
éá:éω1∗=(1,12)§•á´S
ω
1∗
={l
3
}={e
1
e
6
e
14
e
21
}§t
l3
(x(0))=236.00¶gá´
•S
0
ω
1
∗
= {l
5
}={e
4
e
12
e
19
e
23
}§t
l5
(x(0))=279.00"
b
S
ω1
∗
(0)={l
3
,l
5
}"
éO/D:éω2=(3,10)§•á´S
ω
2
= {l
4
}={e
2
e
5
e
11
e
17
}§t
l4
(x(0))=161.00"
b
S
ω2
(0)={l
4
}"
Figure3.Thefigureofnetworkweightingunderx(0)
ã3.3x(0)eä\ã
4)§S(0)={l
3
§l
5
§l
4
}§H(1)={l
1
§l
2
§l
3
§l
4
§l
5
}§Ù¥x
1
(0),x
2
(0),x
3
(0),x
4
(0),x
5
(0)•´
»l
1
,l
2
,l
3
,l
4
,l
5
6þ§=\\´»l
3
§l
4
§l
5
")e5y¯KMP(1):
DOI:10.12677/aam.2019.871401218A^êÆ?Ð
±Œ 
Min z(x) =
P
a∈E
R
x
a
(i)
0
t
a
(x) dx =
R
x
e3
(0)
0
(3x
2
+50) dx +
R
x
e10
(0)
0
(4x
2
+35) dx +
R
x
e18
(0)
0
(3x
2
+
13)dx+
R
x
e23
(0)
0
(3x
2
+90)dx+
R
x
e7
(0)
0
(2x
2
+30)dx++
R
x
e14
(0)
0
(5x
2
+22)dx+
R
x
e20
(0)
0
(3x
2
+60)
dx+
R
x
e22
(0)
0
(2x
2
+70)dx+
R
x
e1
(0)
0
(2x
2
+23)dx+
R
x
e6
(0)
0
(2x
2
+28)dx+
R
x
e21
(0)
0
(4x
2
+38)dx
+
R
x
e2
(0)
0
(x
2
+ 33)dx+
R
x
e5
(0)
0
(3x
2
+ 20)dx+
R
x
e11
(0)
0
(2x
2
+ 42)dx+
R
x
e17
(0)
0
(2x
2
+ 66)dx+
R
x
e4
(0)
0
(2x
2
+11)dx+
R
x
e12
(0)
0
(3x
2
+45)dx+
R
x
e19
(0)
0
(3x
2
+25)dx§
Ù¥(x
e3
(0)=x
e10
(0)=x
e18
(0)=x
1
(0),x
e7
(0)=x
e20
(0)=x
e22
(0)=x
2
(0),x
e1
(0)=
x
e21
(0) = x
e6
(0) = x
3
(0),x
e2
(0) = x
e5
(0) = x
e11
(0) = x
e17
(0) = x
4
(0),x
e4
(0) = x
e12
(0) = x
e19
(0) =
x
5
(0),x
e23
(0) = x
1
(0)+x
5
(0),x
e14
(0) = x
2
(0)+x
3
(0))"
Ïd:Minz(x)=
10
3
(x
1
(0))
3
+188x
1
(0)+(x
1
(0)+x
5
(0))
3
+171x
5
(0)+
7
3
(x
2
(0))
3
+182x
2
(0)+
5
3
(x
2
(0)+x
3
(0))
3
+111x
3
(0)+
8
3
(x
3
(0))
3
+
8
3
(x
4
(0))
3
+161x
4
(0)+
8
3
(x
5
(0))
3
§
s.t





















x
1
(0)+x
3
(0)+x
5
(0) = 6,x
2
(0)+x
4
(0) = 5,
x
1
(0)+x
5
(0) ≤7,x
2
(0)+x
3
(0) ≤6,
0 ≤x
1
(0) ≤7,0 ≤x
2
(0) ≤6,
0 ≤x
3
(0) ≤5,0 ≤x
4
(0) ≤3,
0 ≤x
5
(0) ≤4,
)x(1)3l
1
þ6þx
1
(1)=1.46!3l
2
þ6þx
2
(1)=2.00!3l
3
þ6þx
3
(1)=2.35!
3l
4
þ6þx
4
(1) = 3.00!3l
5
þ6þx
5
(1) = 2.19§Ù§´»þ6þ•0"
6x(1)kÚle
11
§Ú´»l
4
"
36x(1)e§íØÚle
11
§±x(1)•6ˆlþ|G•§\ä
b
ℵ
1
"
i=1§
1¤§(1)éO/D:éω1=(1,12)§T
ω1
(1) = t
l3
(x(1))
.
= 249.50§éO/D:éω2=(3,10)§T
ω2
(1) =
t
l2
(x(1))
.
= 304.43"
(2)36x(1)e§íØÚle
11
§±x(1)•6ˆlþ|G•§\ä
b
ℵ
1
§3\
ä
b
ℵ
1
¥§éO/D:éω1=(1,12)§•á´S
ω
1
= {l
1
}={e
3
e
10
e
18
e
23
}§
e
T
ω1
(1)=t
l1
(x(1))
.
= 249.50¶
éO/D:éω2=(3,10)§•á´S
ω
2
= {l
6
}={e
7
e
13
e
19
e
22
}§
e
T
ω2
(1)=t
l6
(x(1))
.
= 170.40"
Ω
ω1
(1) = 0.00§Ω
ω2
(1) = 134.03"Ω(1) = 134.03"
2¤§ω2∗∈{ω∈W: Ω
ω2
(1) = Ω(1)}§á:é•ω2∗=(3,10)"
éá:éω2∗=(3,10)§•á´•S
ω
2
∗
={l
6
}={e
7
e
13
e
19
e
22
}§t
l6
(x(1))
.
=170.40§gá´
•S
0
ω
2
∗
= {l
7
}={e
2
e
6
e
13
e
19
e
22
}§t
l7
(x(1))
.
= 213.41§
b
S
ω2
∗
(1)={l
6
,l
7
}"
éO/D:éω1=(1,12)§•á´•S
ω
1
={l
1
}={e
3
e
10
e
18
e
23
}§t
l1
(x(1))
.
=249.50§
b
S
ω1
(1)=
{l
1
}"
3¤§S(1)={l
6
§l
7
§l
1
},H(2)={l
1
§l
2
§l
3
§l
4
§l
5
§l
6
§l
7
}§Ù¥x
1
(1),x
2
(1),x
3
(1),x
4
(1),x
5
(1),
x
6
(1),x
7
(1)•´»l
1
,l
2
,l
3
,l
4
,l
5
,l
6
,l
7
6þ§=\\´»l
6
§l
7
")e5y¯KMP(2):
Min z(x) =
P
a∈E
R
x
a
(i)
0
t
a
(x) dx =
R
x
e3
(1)
0
(3x
2
+50) dx +
R
x
e10
(1)
0
(4x
2
+35) dx +
R
x
e18
(1)
0
(3x
2
+
DOI:10.12677/aam.2019.871401219A^êÆ?Ð
±Œ 
13)dx+
R
x
e23
(1)
0
(3x
2
+90)dx+
R
x
e7
(1)
0
(2x
2
+30)dx++
R
x
e14
(1)
0
(5x
2
+22)dx+
R
x
e20
(1)
0
(3x
2
+60)
dx+
R
x
e22
(1)
0
(2x
2
+70)dx+
R
x
e1
(1)
0
(2x
2
+23)dx+
R
x
e6
(1)
0
(2x
2
+28)dx+
R
x
e21
(1)
0
(4x
2
+38)dx
+
R
x
e2
(1)
0
(x
2
+ 33)dx+
R
x
e5
(1)
0
(3x
2
+ 20)dx+
R
x
e11
(1)
0
(2x
2
+ 42)dx+
R
x
e17
(1)
0
(2x
2
+ 66)dx+
R
x
e4
(1)
0
(2x
2
+11)dx+
R
x
e12
(1)
0
(3x
2
+45)dx+
R
x
e19
(1)
0
(3x
2
+25)dx+
R
x
e13
(1)
0
(3x
2
+15)dx§
Ù¥(x
e3
(1)=x
e10
(1)=x
e18
(1)=x
1
(1),x
e23
(1)=x
1
(1)+x
5
(1),x
e7
(1)=x
2
(1)+x
6
(1),
x
e14
(1)=x
2
(1)+x
3
(1),x
e20
(1)=x
2
(1),x
e22
(1)=x
2
(1)+x
6
(1)+x
7
(1),x
e1
(1)=x
e21
(1)=
x
3
(1),x
e6
(1)=x
3
(1)+ x
7
(1),x
e2
(1)=x
4
(1)+ x
7
(1),x
e5
(1)=x
e11
(1)=x
e17
(1)=x
4
(1),x
e4
(1)=
x
e12
(1) = x
5
(1),x
e19
(1) = x
5
(1)+x
6
(1)+x
7
(1),x
e13
(1) = x
6
(1)+x
7
(1))"
Ïd:Minz(x)=
P
a∈E
R
x
a
(i)
0
t
a
(x)dx=
10
3
(x
1
(1))
3
+188x
1
(1)+(x
1
(1)+x
5
(1))
3
+171x
5
(1)+
2
3
(x
2
(1)+x
6
(1))
3
+182x
2
(1)+140x
6
(1)+
5
3
(x
2
(1)+x
3
(1))
3
+111x
3
(1)+(x
2
(1))
3
+
2
3
(x
2
(1)+x
6
(1)+
x
7
(1))
3
+ 171x
7
(1) +2(x
3
(1))
3
+
2
3
(x
3
(1) +x
7
(1))
3
+
1
3
(x
4
(1) +x
7
(1))
3
+ 161x
4
(1) +
7
3
(x
4
(1))
3
+
5
3
(x
5
(1))
3
+(x
5
(1)+x
6
(1)+x
7
(1))
3
+(x
6
(1)+x
7
(1))
3
§
s.t



























x
1
(1)+x
3
(1)+x
5
(1) = 6,x
2
(1)+x
4
(1)+x
6
(1)+x
7
(1) = 5,
x
1
(1)+x
5
(1) ≤7,x
2
(1)+x
3
(1) ≤6,x
2
(1)+x
6
(1)+x
7
(1) ≤8,
x
2
(1)+x
6
(1) ≤6,x
3
(1)+x
7
(1) ≤5,x
5
(1)+x
6
(1)+x
7
(1) ≤4,
x
6
(1)+x
7
(1) ≤6,x
4
(1)+x
7
(1) ≤6,0 ≤x
1
(1) ≤7,
0 ≤x
2
(1) ≤6,0 ≤x
3
(1) ≤5,0 ≤x
4
(1) ≤3,0 ≤x
5
(1) ≤5,
0 ≤x
6
(1) ≤4,0 ≤x
7
(1) ≤4.
)x(2)3l
1
þ6þx
1
(2)=1.67!3l
2
þ6þx
2
(2)=0.00!3l
3
þ6þx
3
(2)=3.15!
3l
4
þ6þx
4
(2)=2.64!3l
5
þ6þx
5
(2)=1.18!3l
6
þ6þx
6
(2)=2.36!3l
7
þ6
þx
7
(2) = 0.00§Ù§´»þ6þ•0"
6x(2)vkÚl§±x(2)•6ˆlþ|G•§\ä
b
ℵ
2
"
i=2§
1¤§£1¤éO/D:éω1=(1,12)!T
ω1
(2) = t
l3
(x(2))
.
= 240.08§éO/D:éω2=(3,10)§T
ω2
(2)) =
t
l6
(x(2))
.
= 216.70"
£2¤3\ä
b
ℵ
2
¥§éO/D:éω1=(1,12)§•á´S
ω
1
={l
8
}={e
4
e
11
e
18
e
23
}§
e
T
ω1
(2)
=t
l8
(x(2))
.
= 205.40¶éO/D:éω2=(3,10)§•á´S
ω
2
= {l
4
}={e
2
e
5
e
11
e
17
}§
e
T
ω 2
(2)=t
l4
(x(2))
.
=
216.70"
Ω
ω1
(2) = 34.68§Ω
ω2
(2) = 0.00"Ω(2) = 34.68"
2¤§ω1∗∈{ω∈W: Ω
ω1
(2) = Ω(2)}§á:é•ω1∗=(1,12)"
3\ä
b
ℵ
2
¥§éO/D:éω1∗=(1,12)§T
ω1
(2)=t
l3
(x(2))
.
=240.08§éO/D:éω2
=(3,10)§
T
ω2
(2) = t
l6
(x(2))
.
= 216.70"
Œ16x(2)á•Ω(2) = 34.68§á:é•ω1∗=(1,12)"
éá:éω1∗=(1,12)§•á´•S
ω
1
∗
={l
8
}={e
4
e
11
e
18
e
23
}§t
l8
(x(2))
.
=205.40§gá´
DOI:10.12677/aam.2019.871401220A^êÆ?Ð
±Œ 
•S
0
ω
1
∗
= {l
5
}={e
4
e
12
e
19
e
23
}§t
l5
(x(2))
.
= 240.06"
b
S
ω1∗
(2)={l
8
,l
5
}"
éO/D:éω2=(3,10),•á´•S
ω
2
(2)={l
4
}={e
2
e
5
e
11
e
17
}§t
l4
(x(2))
.
=216.70"
b
S
ω2
(2)=
{l
4
}"
3¤§S(2) ={l
8
§l
5
§l
4
}§H(3)={l
1
§l
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x
a
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0
t
a
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7
3
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1
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3
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1
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8
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3
+156x
8
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1
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x
5
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8
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3
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3
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3
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(x
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3
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2
3
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6
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7
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3
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7
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3
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3
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2
3
(x
3
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7
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3
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1
3
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4
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7
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3
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161x
4
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5
3
(x
4
(2))
3
+
2
3
(x
4
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8
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3
+
2
3
(x
5
(2)+x
8
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3
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5
(2))
3
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5
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6
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7
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3
+
(x
6
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7
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3
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s.t
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
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x
1
(2)+x
3
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5
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8
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2
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6
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7
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x
1
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8
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1
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5
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8
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2
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6
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x
3
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2
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2
(2)+x
6
(2)+x
7
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3
(2)+x
7
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x
4
(2)+x
7
(2) ≤6,x
4
(2)+x
8
(2) ≤3,x
5
(2)+x
8
(2) ≤5,
x
5
(2)+x
6
(2)+x
7
(2) ≤4,x
6
(2)+x
7
(2) ≤6,0 ≤x
1
(2) ≤7,
0 ≤x
2
(2) ≤6,0 ≤x
3
(2) ≤5,0 ≤x
4
(2) ≤3,0 ≤x
5
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0 ≤x
6
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7
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8
(2) ≤3.
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1
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1
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8
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e23
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1
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5
(2)+x
8
(2),x
e7
(2) =
x
2
(2) + x
6
(2),x
e14
(2)=x
2
(2) + x
3
(2),x
e20
(2)=x
2
(2),x
e22
(2)=x
2
(2) + x
6
(2) + x
7
(2),x
e1
(2)=
x
e21
(2)=x
3
(2),x
e6
(2)=x
3
(2)+x
7
(2),x
e2
(2)=x
4
(2)+x
7
(2),x
e5
(2)=x
e17
(2)=x
4
(2),x
e11
(2)=
x
4
(2) + x
8
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e4
(2)=x
5
(2) + x
8
(2),x
e12
(2)=x
5
(2),x
e19
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5
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6
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7
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e13
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x
6
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1
þ6þx
1
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3
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3
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3l
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=238.90"éO/D:éω2=(3,10)§T
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6
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.
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e
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l
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19
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22
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l
6
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.
=230.90"
DOI:10.12677/aam.2019.871401221A^êÆ?Ð
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ë•©z
[1]Wardrop,J.(1952)SomeTheoreticalAspectsofRoadTrafficResearch.Proceedingsofthe
InstituteofCivilEngineers,PartII,1,325-378.https://doi.org/10.1680/ipeds.1952.11259
[2]Beckmann, M.J.,McGuire,C.B. andWinsten,C.B. (1956)Studiesin theEconomics ofTrans-
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[3]Lin,Z.(2010)TheStudyofTrafficEquilibriumProblemswithCapacityConstraintsofArcs.
NonlinearAnalysis:RealWorldApplications,11,2280-2284.
https://doi.org/10.1016/j.nonrwa.2009.07.002
[4]Lin,Z.(2010)OnExistenceofVectorEquilibriumFlowswithCapacityConstraintsofArcs.
NonlinearAnalysis:Theory,Methods&Applications,72,2076-2079.
https://doi.org/10.1016/j.na.2009.10.007
[5]Lin,Z.(2015)AnAlgorithmforTrafficEquilibriumFlowwithCapacityConstraintsofArcs.
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[6]Xu,Y.D. and Li, S.J. (2014) Vector Network Equilibrium Problems with Capacity Constraints
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https://doi.org/10.1080/00036811.2013.875160
[7]Tian,X.Q.andXu,Y.D.(2012)TrafficNetworkEquilibriumProblemswithCapacityCon-
straintsofArcsandLinearScalarizationMethods.JournalofAppliedMathematics,2012,
ArticleID:612142.https://doi.org/10.1155/2012/612142
[8]Xu,Y.D.,Li,S.J.andTeo,K.L.(2012)VectorNetworkEquilibriumProblemswithCapacity
Constraints ofArcs. TransportationResearchPartE:LogisticsandTransportationReview, 48,
567-577.https://doi.org/10.1016/j.tre.2011.11.002
[9]Chiou,S.W.(2010)AnEfficientAlgorithmforComputingTrafficEquilibriaUsingTransyt
Model.AppliedMathematicalModelling,34,3390-3399.
https://doi.org/10.1016/j.apm.2010.02.028
[10]Xu,M.,Chen,A.,Qu,Y.andGao,Z.(2011)ASemismoothNewtonMethodforTraffic
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[11]Chen,A.,Zhou,Z.andXu,X.D.(2012)ASelf-AdaptiveGradientProjectionAlgorithmfor
the Nonadditive Traffic Equilibrium Problem. Computers&OperationsResearch, 39, 127-138.
https://doi.org/10.1016/j.cor.2011.02.018
DOI:10.12677/aam.2019.871401223A^êÆ?Ð
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