In this section, the proposed approach is applied to two more complicated networks for demonstrating its feasibility and availability. All algorithms are implemented using MATLAB.

#### 5.2. Case Study

In this numerical experiment, a larger size real-world network is selected to demonstrate the proposed approach. The real traffic network (left upper corner in

Figure 5) is located at the H-tech district of Xi’an in China and has 49 nodes and 150 links. Except for links 26, 53 and 58 that are one-way links, the others have bi-directional flows between two nodes (as shown in

Figure 5). The reason to consider the main routes is that parking is not permitted in many zones and users choose alternative routes as practical as possible rather than randomly. Therefore, in this real network, we consider 176 O-D pairs and 347 routes. It’s a huge budget for a large traffic network to obtain full observability only considering a camera system. In other words, the budget is a critical constraint for extracting sufficient flow information. To illustrate this problem and determine full routes flow, the bi-level programming model as indicated in the previous section is applied to this case study.

The following set of selected links with single scanning sensors is obtained from the first level model and algorithm after cpu 0.461s.

$U$ = {118, 101, 109, 41, 108, 98, 107, 115, 124, 50, 69, 140, 133, 27, 125, 14, 19, 80, 75, 84, 10, 104, 79, 70, 150, 49, 23, 147, 76, 97, 35, 120, 32, 148, 110, 99, 38, 119, 36, 149, 31, 131, 26, 96, 60, 139, 141, 6, 29, 100, 117, 77, 136, 134, 44, 5, 59, 73, 8, 24 ,33, 113, 34, 114, 144, 43, 78, 12, 103 ,137, 11, 4, 13, 39, 20, 81, 123, 132}

When only scanning sensors are considered to extract all routes flow, the near-optimal solution is the set of 79 scanned links, which indicates 53% of the links to be scanned. Compared to the previous in the Cuenca network (the minimum 34% (175) scanned links sufficient for full observability) [

3], the result is relatively higher because of the different network topology and route. The truth is important for a much larger network to estimate the cost of full observability. However, accounting for the high cost of scanning sensors and demand of optimizing the budget, the second level model and algorithm are implemented in the real network.

In this real network, the maximum of replaced scanning sensors is equal to 21 based on the result of second level algorithm. The near optimal combination scheme is shown as follows:

${U}_{s}$ = {118, 101, 109, 41, 108, 98, 107, 115, 124, 50, 69, 140, 133, 27, 125, 14, 19, 80, 75, 84, 10, 104, 79, 150, 49, 147, 76, 97, 35, 120, 32, 148, 110, 38, 119, 36, 149, 31, 131, 139, 141, 6, 100, 77 ,136, 134, 59, 8, 24 ,33, 113, 34, 11, 4, 20, 81, 132}

${U}_{c}=\{$123, 39, 13, 137, 103, 12, 78, 43, 144, 114, 73, 5, 44, 117, 29, 60, 26, 96, 99, 23, 70}

This scheme consists of 56 scanning sensors and 21 counting sensors. It’s necessary to emphasize this scheme is a relatively optimal feasible solution to observe all routes or routes of interest flow. The feasible schemes are far more than one, especially for a larger traffic network. The computational times increase exponentially with the increase of scanned links in the first level algorithm, therefore this paper only focuses on obtaining one near optimal solution in the real traffic network.

To illustrate the total cost of different combinations, inductive loop and video detection systems are chosen to represent counting and scanning sensors, respectively. The price of an inductive loop varies from

$600 to

$900 and the price of a video detection system varies from

$2400 to

$6000 [

12]. Assume

${c}_{1}$ and

${c}_{2}$ are the prices of a video detection system and inductive loop, respectively, then

${c}_{1}/{c}_{2}$ approximately varies from 3 to 10.

${n}_{ws}$ and

${n}_{wc}$ is the number of video detection systems and inductive loops in scheme

$w$, respectively. As shown in

Figure 6, the replacement number refers to the different combination schemes and the maximum value is equal to 21.

The total cost of different combination schemes can be calculated by the following formula:

As shown in

Figure 6, the total cost is monotonously increasing in the horizontal and longitudinal direction. In other words, the total cost decreases as the number of replaced sensors increases and rises with the growth of ratio of

${c}_{1}$ to

${c}_{2}$. The most economical combination scheme is composed of 56 scanning sensors and 21 counting sensors, and its total cost is equal to 195 units (

${c}_{2}$). The deployment scheme of 79 scanning sensors, namely a single type of sensors, is the most expensive and the total cost reaches up to 790 unit (

${c}_{2}$). The maximum cost is approximately four times higher than the minimum, as shown in the cost distribution diagram. In other words, the total cost of sensors layout for full flow observability is reduced by 75% through implementing the proposed approach in this case study. There exists different reductions because of different network topologies and scales. Obviously, the combination deployment scheme outperforms the scheme using a single type of sensors from a viewpoint of sensor deployment cost. Unfortunately, few previous studies paid attention to combine the multi-type of sensors for observability in their case studies. Therefore, it’s not trivial to optimize and combine different types of sensors to implement full route observability, especially for a larger network.

It’s necessary to indicate that the proposed approach requires the information of all or interesting routes. It’s critical for the output result that extracting routes information be as practical as possible before generating an input matrix. Again, one feasible near-optimal solution can be obtained from the proposed approach because it is computationally NP-hard. The proposed approach provides many different combination schemes that consist of different types and number of sensors. Considering the maintenance and stability of sensors, the transportation agency can replace old counting sensors with scanning sensors stepwise according to budget constraints, which give lower maintenance costs and more information.