The equilibrium network signal control problem is represented as a Stackelberg game. Due to the characteristics of a Stackelberg game, solving the upper-level problem and lower-level problem iteratively cannot be expected to converge to the solution. The reaction function of the lower-level problem is the key information to solve a Stackelberg game. Usually, the reaction function is approximated by the network sensitivity information. This paper firstly presents the general form of the second-order sensitivity formula for equilibrium network flows. The second-order sensitivity information can be applied to the second-order reaction function to solve the network signal control problem efficiently. Finally, this paper also demonstrates two numerical examples that show the computation of second-order sensitivity and the speed of convergence of the nonlinear approximation algorithm.
The network signal control problem (NSCP) is to find the optimal signal setting which improves the performance of existing facilities in a transportation network. Conventional methods for optimizing signal settings can be divided into two types: stage-based and group-based approaches [
Marcotte [
The remainder of this paper is organized as follows: Section
Consider a transportation network consisting of a finite set of nodes
In sensitivity analysis, a vector of perturbation parameters with dimension
An equivalent VI can be written with the cost function in terms of path flow variable
The classical first-order sensitivity analysis for equilibrium network flows was proposed by Tobin and Friesz [
In the row reduction method, a maximal set of rows from
The sensitivity analysis-based nonlinear approximation heuristic algorithm (NLAA) was firstly proposed by Cho and Lin [
To derive the second-order sensitivity formula, we introduce some definitions and theorems of matrix calculus as follows [
Let
Let
Let
Let
Let
Let
To derive the second-order sensitivity formula for equilibrium network flows, it is intuitive to take derivative of (
The second-order sensitivity for equilibrium network flows is
Since the first-order sensitivity is the product of (
Consider the signal optimization problem, where the aim of the regulating agency is to minimize a network performance function
In the general problem, the signal variable that can be set by the controlling agent is green time. By specifying the cost functions
If
The iterative optimization assignment (IOA) method described by Tan et al. [
The challenge in solving problem
The heuristic is detailed as follows.
Determine a fixed small value
Solve (
Calculate the sensitivity information
Using
Reformulate (
Solve the problem in Step
In the sensitivity analysis-based linear approximation heuristic algorithm, the reaction function of the lower level is based on approximation by a linear function. In this section, the reaction function of the lower-level problem is based on approximation by a nonlinear function and is plugged into the upper-level problem and is iterated until the solutions converge (abbreviated as NLAA) [
Determine a fixed small value
Solve (
Calculate the sensitivity information
Using
Reformulate (
Solve the problem in Step
In addition to describing the algorithm in more detail, we will provide a proof that if this algorithm converges, it converges to an optimal solution of problem
If algorithm A2 converges, it converges to a critical point of
If the sequence if we set let
Then, by the Karush-Kuhn-Tucker necessary conditions for optimality of vectors
So, taking the derivative with respect to
This section provides two numerical examples which illustrate the computation of second-order sensitivity and the performance of NLAA. The first example demonstrates the computation of second-order sensitivity in detail. The second example focuses on the speed of convergence between LAA and NLAA.
The first example is chosen from Dickson [
The network topology in Example
Additionally, the travel demand
Arc cost functions and the system objective function in Example
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Arc number |
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1 | 2 | 1 |
2 | 0 | 2 |
3 | 0 | 2 |
In this example,
In (
From Lemma
In this example, the matrix
Based on (
In this example, both LAA and NLAA are implemented in the MATLAB environment. Set
Computational results of LAA and NLAA in Example
Iteration | LAA | NLAA | ||||
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1 | 8.44693 | 7.63647 | 47.23792 | 8.45125 | 7.70052 | 47.23576 |
2 | 8.45366 | 7.73667 | 47.23553 | 8.45326 | 7.73055 | 47.23552 |
3 | 8.45323 | 7.73019 | 47.23552 | 8.45326 | 7.73056 | 47.23552 |
4 | 8.45325 | 7.73040 | 47.23552 |
This example is a simplified real network which represents the afternoon rush hour traffic between the working area Hsinchu Science-Based Industrial Park (HSIP) and the residential area Jhubei city. The network topology follows Figure
In this example, we set
Arc cost functions and the system objective function in Example
Signalized arc cost function |
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Nonsignalized arc cost function |
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System objective function |
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Arc number |
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1 | 1.8545 | 0.9200 | 3.5800 | 56.6667 | 300 |
2 | 0.8667 | 0.9200 | 3.5800 | 40.0000 | — |
3 | 1.8000 | 1.2700 | 3.9600 | 115.0000 | — |
4 | 2.2364 | 1.2700 | 3.9600 | 115.0000 | — |
5 | 0.2945 | 1.2100 | 2.3900 | 28.3333 | 300 |
6 | 0.1964 | 1.4200 | 2.3200 | 85.0000 | 300 |
7 | 0.3818 | 0.8600 | 4.3400 | 85.0000 | 300 |
8 | 1.0154 | 1.2700 | 3.9600 | 68.3333 | — |
9 | 1.0000 | 1.2100 | 2.3900 | 20.0000 | — |
10 | 1.0154 | 1.2700 | 3.9600 | 68.3333 | 180 |
11 | 0.3273 | 0.9200 | 3.5800 | 56.6667 | 180 |
12 | 0.9818 | 1.4200 | 2.3200 | 85.0000 | 150 |
13 | 0.6545 | 0.8600 | 4.3400 | 113.3333 | 150 |
14 | 1.2000 | 1.5000 | 2.4400 | 113.3333 | 150 |
15 | 3.8727 | 1.2700 | 3.9600 | 115.0000 | — |
16 | 0.4909 | 0.9200 | 3.5800 | 40.0000 | 150 |
Origin-destination demand table in Example
Destination | ||||||||
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4 | 6 | 8 | 11 | 13 | 14 | 16 | ||
Origin | 1 | 50 | 275 | 475 | 400 | 1250 | 275 | 2250 |
2 | 0 | 0 | 0 | 0 | 2550 | 0 | 1400 | |
5 | 0 | 150 | 250 | 200 | 0 | 150 | 250 | |
7 | 0 | 0 | 500 | 400 | 0 | 300 | 450 | |
9 | 0 | 0 | 0 | 325 | 0 | 225 | 350 | |
12 | 0 | 0 | 0 | 0 | 0 | 125 | 175 | |
15 | 0 | 0 | 0 | 0 | 0 | 0 | 900 |
Computational results of LAA and NLAA in Example
Arc number | LAA | NLAA | ||||
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1 | 195.9103 | 31.4891 | 2.8118 | 195.8881 | 31.4901 | 2.8123 |
2 | — | 51.4275 | 2.8272 | — | 51.4266 | 2.8271 |
3 | — | 65.8333 | 2.0510 | — | 65.8333 | 2.0510 |
4 | — | 117.2609 | 5.3043 | — | 117.2599 | 5.3042 |
5 | 104.0897 | 16.6667 | 1.5529 | 104.1119 | 16.6667 | 1.5522 |
6 | 181.5056 | 47.3225 | 0.4263 | 181.4711 | 47.3234 | 0.4265 |
7 | 118.4944 | 27.5000 | 0.5199 | 118.5289 | 27.5000 | 0.5197 |
8 | — | 67.7391 | 2.2611 | — | 67.7401 | 2.2612 |
9 | — | 15.0000 | 1.6084 | — | 15.0000 | 1.6084 |
10 | 167.0451 | 62.3225 | 2.2192 | 167.0853 | 62.3234 | 2.2181 |
11 | 12.9549 | 5.0000 | 0.9517 | 12.9147 | 5.0000 | 0.9587 |
12 | 123.8799 | 45.2391 | 1.4849 | 123.8808 | 45.2401 | 1.4849 |
13 | 26.1201 | 15.0000 | 0.8256 | 26.1192 | 15.0000 | 0.8256 |
14 | 45.4083 | 42.3225 | 4.2042 | 45.4092 | 42.3234 | 4.2042 |
15 | — | 63.3333 | 4.3361 | — | 63.3333 | 4.3361 |
16 | 104.5917 | 53.9275 | 5.2760 | 104.5908 | 53.9266 | 5.2759 |
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Iteration number | 101 | 6 | ||||
Objective value ( |
2188.2886 | 2188.2404 |
The network topology in Example
The convergence curves of LAA and NLAA in Example
The key information to solve the equilibrium network signal control problem (ENSCP) is the reaction function of the lower-level problem. Because the reaction function cannot be obtained explicitly, the sensitivity information of equilibrium network flows is used to approximate it. Based on the first-order sensitivity formula and the matrix calculus, this paper first presents the general form of the second-order sensitivity formula for equilibrium network flows. With the second-order sensitivity formula, the reaction function can be approximated more accurately by a nonlinear function. From HSIP to Jhubei city, a simplified real network example demonstrates the speed of convergence between LAA and NLAA. The NLAA has significant improvement in solving the ENSCP with complicated arc cost functions; in this example, the NLAA only takes 6% iterations to attain the same level of precision.
This study focuses on the NLAA and a simplified delay formula is adopted to reflect the influence of traffic congestion. Practically, a traffic propagation model, such as TRANSYT model, should be included when solving the ENSCP. Since the derivatives of TRANSYT model have been obtained explicitly [
Compared with LAA, the number of multiplications for matrix multiplication is greatly increasing in NLAA due to the Kronecker-product operation. NLAA has polynomial complexity with the network size and the number of perturbation parameters because of the property of the Kronecker product. There is still opportunity to improve the computing efficiency through adopting effective Kronecker-product algorithms.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was partially supported by the National Science Council, Taiwan; Ministry of Transportation and Communications, Taiwan; Chunghwa Telecom Laboratories, Taiwan, under contract number NSC-100-2221-E-009-119, MOTC-STAO-101-02, MOTC-STAO-102-02, TL-103-G109, respectively. The authors also acknowledge the reviewer’s good suggestions for the further research topics.