Consider a freeway segment with four on-ramps (and the corresponding four sections) as shown in the figure below. Demand at input i, Di, (i = 1, ..., 5), capacity at section j, B; (i = 1, ..., 4), proportion of vehicles entering at input i which pass through section j, Aij (i = 1, ..., 5; j = 1, ..., 4), and minimum ramp metering rates at input i, Xmin, i (i =2, ..., 5) are provided in the tables below. D₁ = 4500 vph Section 1 B₁ = 5200 vph S1 Section 2 B₂ = 5010 vph S₂ Section 3 B₁ = 5100 vph Section 4 B₁ = 5300 vph S3 S4 ramp 1 ramp 2 ramp 3 ramp 4 Ха D5= 700 vph D4 = 750 vph X, D₂ = 850 vph D3 = 550 vph Capacity at section j (B;) Demand at input i (D) Demand Di (vph) 4500 Input i 1 2 850 3 550 4 750 5 700 Let X denote allowable ramp metering rates at input i (i=2, 3, 4, and 5). With the objective of maximizing ramp input flows and with the considerations of the downstream demand-capacity constraints and the minimum ramp metering rates, 1. Determine the ramp metering rates X2, X3, X4, X5 [16 pts]. 2. Assuming green time is 3 seconds. What would be the red time for ramp 2 and ramp 5 [9 pts]. Objective function: Max X (where n the number of inputs) Section j Capacity B; (vph) Subject to: S=AX ≤B (j = 1,...,n-1) 1 5200 X≤D (i=2,...,n) 2 5010 XXmini (i = 2,...,n) 3 5100 X₁ ≤ Xmaxi (i = 2,...,n) 4 5300 Proportion of vehicles entering at input i which pass-through section j (Aij) Minimum Ramp Metering Rates (Xmin, i) j 1 2 3 4 Ramp i Xmin, i (vph) i 2 240 1 1.00 0.95 0.90 0.85 3 240 2 1.00 0.75 0.70 0.60 4 3 1.00 0.90 0.85 240 4 1.00 0.90 5 240 5 1.00 Constraint (1) Constraint (2) Constraint (3) Constraint (4)
Consider a freeway segment with four on-ramps (and the corresponding four sections) as shown in the figure below. Demand at input i, Di, (i = 1, ..., 5), capacity at section j, B; (i = 1, ..., 4), proportion of vehicles entering at input i which pass through section j, Aij (i = 1, ..., 5; j = 1, ..., 4), and minimum ramp metering rates at input i, Xmin, i (i =2, ..., 5) are provided in the tables below. D₁ = 4500 vph Section 1 B₁ = 5200 vph S1 Section 2 B₂ = 5010 vph S₂ Section 3 B₁ = 5100 vph Section 4 B₁ = 5300 vph S3 S4 ramp 1 ramp 2 ramp 3 ramp 4 Ха D5= 700 vph D4 = 750 vph X, D₂ = 850 vph D3 = 550 vph Capacity at section j (B;) Demand at input i (D) Demand Di (vph) 4500 Input i 1 2 850 3 550 4 750 5 700 Let X denote allowable ramp metering rates at input i (i=2, 3, 4, and 5). With the objective of maximizing ramp input flows and with the considerations of the downstream demand-capacity constraints and the minimum ramp metering rates, 1. Determine the ramp metering rates X2, X3, X4, X5 [16 pts]. 2. Assuming green time is 3 seconds. What would be the red time for ramp 2 and ramp 5 [9 pts]. Objective function: Max X (where n the number of inputs) Section j Capacity B; (vph) Subject to: S=AX ≤B (j = 1,...,n-1) 1 5200 X≤D (i=2,...,n) 2 5010 XXmini (i = 2,...,n) 3 5100 X₁ ≤ Xmaxi (i = 2,...,n) 4 5300 Proportion of vehicles entering at input i which pass-through section j (Aij) Minimum Ramp Metering Rates (Xmin, i) j 1 2 3 4 Ramp i Xmin, i (vph) i 2 240 1 1.00 0.95 0.90 0.85 3 240 2 1.00 0.75 0.70 0.60 4 3 1.00 0.90 0.85 240 4 1.00 0.90 5 240 5 1.00 Constraint (1) Constraint (2) Constraint (3) Constraint (4)
Traffic and Highway Engineering
5th Edition
ISBN:9781305156241
Author:Garber, Nicholas J.
Publisher:Garber, Nicholas J.
Chapter9: Capacity And Level Of Service For Highway Segments
Section: Chapter Questions
Problem 7P
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Please explain step by step and show formula
![Consider a freeway segment with four on-ramps (and the corresponding four sections) as shown in the
figure below. Demand at input i, Di, (i = 1, ..., 5), capacity at section j, B; (i = 1, ..., 4), proportion of
vehicles entering at input i which pass through section j, Aij (i = 1, ..., 5; j = 1, ..., 4), and minimum ramp
metering rates at input i, Xmin, i (i =2, ..., 5) are provided in the tables below.
D₁ = 4500 vph
Section 1
B₁ = 5200 vph
S1
Section 2
B₂ = 5010 vph
S₂
Section 3
B₁ = 5100 vph
Section 4
B₁ = 5300 vph
S3
S4
ramp 1
ramp 2
ramp 3
ramp 4
Ха
D5= 700 vph
D4 = 750 vph
X,
D₂ = 850 vph
D3 = 550 vph
Capacity at section j (B;)
Demand at input i (D)
Demand Di (vph)
4500
Input i
1
2
850
3
550
4
750
5
700
Let X denote allowable ramp metering rates at input i (i=2, 3, 4, and 5). With the objective of maximizing
ramp input flows and with the considerations of the downstream demand-capacity
constraints and the minimum ramp metering rates,
1. Determine the ramp metering rates X2, X3, X4, X5 [16 pts].
2. Assuming green time is 3 seconds. What would be the red time for ramp 2 and ramp 5 [9 pts].
Objective function: Max X
(where n the number of inputs)
Section j
Capacity B; (vph)
Subject to:
S=AX ≤B (j = 1,...,n-1)
1
5200
X≤D (i=2,...,n)
2
5010
XXmini (i = 2,...,n)
3
5100
X₁ ≤ Xmaxi (i = 2,...,n)
4
5300
Proportion of vehicles entering at input i
which pass-through section j (Aij)
Minimum Ramp
Metering Rates (Xmin, i)
j
1
2
3
4
Ramp i
Xmin, i (vph)
i
2
240
1
1.00
0.95
0.90
0.85
3
240
2
1.00
0.75
0.70
0.60
4
3
1.00
0.90
0.85
240
4
1.00
0.90
5
240
5
1.00
Constraint (1)
Constraint (2)
Constraint (3)
Constraint (4)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F553c03f6-c199-4870-a061-8ddf7adde455%2Feef54f7b-3c69-4147-8752-7ccb037b2e04%2Fndxoxs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a freeway segment with four on-ramps (and the corresponding four sections) as shown in the
figure below. Demand at input i, Di, (i = 1, ..., 5), capacity at section j, B; (i = 1, ..., 4), proportion of
vehicles entering at input i which pass through section j, Aij (i = 1, ..., 5; j = 1, ..., 4), and minimum ramp
metering rates at input i, Xmin, i (i =2, ..., 5) are provided in the tables below.
D₁ = 4500 vph
Section 1
B₁ = 5200 vph
S1
Section 2
B₂ = 5010 vph
S₂
Section 3
B₁ = 5100 vph
Section 4
B₁ = 5300 vph
S3
S4
ramp 1
ramp 2
ramp 3
ramp 4
Ха
D5= 700 vph
D4 = 750 vph
X,
D₂ = 850 vph
D3 = 550 vph
Capacity at section j (B;)
Demand at input i (D)
Demand Di (vph)
4500
Input i
1
2
850
3
550
4
750
5
700
Let X denote allowable ramp metering rates at input i (i=2, 3, 4, and 5). With the objective of maximizing
ramp input flows and with the considerations of the downstream demand-capacity
constraints and the minimum ramp metering rates,
1. Determine the ramp metering rates X2, X3, X4, X5 [16 pts].
2. Assuming green time is 3 seconds. What would be the red time for ramp 2 and ramp 5 [9 pts].
Objective function: Max X
(where n the number of inputs)
Section j
Capacity B; (vph)
Subject to:
S=AX ≤B (j = 1,...,n-1)
1
5200
X≤D (i=2,...,n)
2
5010
XXmini (i = 2,...,n)
3
5100
X₁ ≤ Xmaxi (i = 2,...,n)
4
5300
Proportion of vehicles entering at input i
which pass-through section j (Aij)
Minimum Ramp
Metering Rates (Xmin, i)
j
1
2
3
4
Ramp i
Xmin, i (vph)
i
2
240
1
1.00
0.95
0.90
0.85
3
240
2
1.00
0.75
0.70
0.60
4
3
1.00
0.90
0.85
240
4
1.00
0.90
5
240
5
1.00
Constraint (1)
Constraint (2)
Constraint (3)
Constraint (4)
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