^{1}

^{2}

^{1}

^{1}

^{2}

There are differences between the requirements for traffic network for traffic demand in daily and emergency situations. In order to evaluate how the network designed for daily needs can meet the surging demand for emergency evacuation, the concept of emergency reliability and corresponding evaluation method is proposed. This paper constructs a bilevel programming model to describe the proposed problem. The upper level problem takes the maximum reserve capacity multiplier as the optimization objective and considers the influence of reversible lane measures taken under emergency conditions. The lower level model adopts the combined traffic distribution/assignment model with capacity limits, to describe evacuees’ path and shelter choice behavior under emergency conditions and take into account the traits of crowded traffic. An iterative optimization method is proposed to solve the upper level model, and the lower level model is transformed into a UE assignment problem with capacity limits over a network of multiple origins and single destination, by adding a dummy node and several dummy links in the network. Then a dynamic penalty function algorithm is used to solve the problem. In the end, numerical studies and results are provided to demonstrate the rationality of the proposed model and feasibility of the proposed solution algorithms.

In recent decades, the frequent occurrence of emergencies around the world has caused certain casualties and property losses, posing a great challenge to public security. Especially when these emergencies occur in densely populated urban areas, they are undoubtedly becoming a burden to the already crowded urban traffic. This premise puts forward higher requirements for traffic network under emergency.

Apparently, different from the daily demand of the traffic network, the traffic demand under emergency conditions has an evident feature of unidirection and will surge within a short time. As a result, the existing capacity of road infrastructures is usually not able to meet the demand of evacuation.

To investigate the maximum emergency evacuation demand that can be satisfied by the network designed for daily needs, this paper analyzes the traffic demand under emergency conditions and takes the emergency reliability of traffic network as the research target.

Meanwhile, so as to take full advantage of the finite road resources, ease the traffic pressure which exceeds the load of the road network, increase the evacuation capacity of the road network, and improve the evacuation efficiency, traffic managers will take certain emergency traffic management measures, such as lane reversal. Hence, when studying the traffic network emergency reliability, we should not only consider the particularity of emergency traffic demand, but also consider the impact of these measures on the network service capacity at the same time. As a result, the network under this situation can be deemed as a variable one with capacity constraints, which is more in line with the traits of traffic supply and demand in an emergency.

The remainder of this paper is organized as follows: Section

Traffic reliability research began in the 1980s. Over the past decade, transportation reliability has been a new avenue for advancing transportation research in terms of both theories and practical applications, and has attracted tremendous effort over the past few years [

Chang [

Wael [

Researches in capacity reliability mainly concern about the max-flow of the network in early stage, especially in transportation network. Capacity reliability is defined as the largest multiplier applied to a given existing (or basic) OD demand matrix that can be allocated to a network without violating the link capacities [

The reliability of road network under extreme conditions is very important. In the case of not considering the road damage, how much margin a network has to accommodate emergency traffic demand is an important question that needs to be considered in traffic network design. Based on capacity reliability, this paper evaluates traffic network reliability under emergency conditions from planning perspective.

This section first analyzes the characteristics of evacuation traffic flow, then summarizes emergency traffic management measures under emergency conditions, and finally leads to the establishment of emergency reliability concept.

Different from the traffic state under daily conditions, the traffic state under emergency evacuation has the characteristics of suddenness, high risk, contingency, and so on. In general, emergency evacuation traffic has the following characteristics.

Depending on traffic flow characteristics under emergency conditions, there are several emergency traffic management measures that are commonly used in practice. The purpose is to improve evacuation efficiency and reduce casualties and property losses.

The most widely used emergency traffic control strategy is lane reversal.

Given a traffic network which is designed for daily traffic demand, in order to evaluate the extent how the network can meet the evacuation requirements under emergency conditions, the concept of emergency reliability is proposed here. In this study, the concept of network reserve capacity is used to evaluate emergency reliability of traffic network. Network reserve capacity refers to the maximum traffic demand multiplier that can be applied to the given existing (or basic) traffic OD matrix in the process of trip assignment when the road capacity constraints can be satisfied [

In this paper, traffic demand multiplier

With the constant adjustment of traffic demand multiplier

In the process of trip assignment, it is assumed that the evacuation demand from each origin node is known, and the evacuation demand to each destination depends on the choice of evacuees. Each evacuation destination corresponds to a shelter. And the evacuation demand that each shelter can accommodate is limited. It is assumed that, in case of emergency evacuation, evacuees tend to choose the fastest routes and the nearest shelter in order to reach the safe area as soon as possible, considering the congestion on the routes and in the shelters.

In this section, the mathematical description for the problem of interest is formally proposed with a bilevel programming model to analyze the aforementioned multiplier

Given an existing traffic network

The upper level model aims to maximum the variable

In this model, the total number of lanes of a certain segment,

The upper level model influences the lower level model through two sets of variables. The variable

The lower level model is a traveler behavior model describing route and shelter choices of evacuees in emergency situations. The total number of trips generated at origin nodes is given, and the capacity of a single lane and each shelter is also known. Evacuees will choose the shelter that they can arrive as soon as possible and choose the route by which they can reach the safe area in the shortest time, with the consideration of congestion influence on the routes and in the shelters under emergency circumstance. The number of evacuees reaching each shelter is finally decided by the lower level model. Therefore the lower level model is formulated as a joint UE distribution/assignment model with capacity constraints:

The definitional constraints are as follows:

In this model, (

The objective function of program (

To demonstrate the equivalence conditions of lower level model established above, it has to be shown that any flow pattern that solves the mathematical program (

The Lagrangian of the equivalent minimization problem with respect to all the equality and inequality constraints can be formulated as

According to Karush-Kuhn-Tucker (KKT) conditions, Lagrangian function (

One can solve the partial derivatives in expression (

Here,

Equations (

if

if

That is to say, the paths connecting any O-D pair can be divided into two categories: those carrying flow, on which the generalized travel time equals the minimum O-D generalized travel time; and those not carrying flow, on which the generalized travel time is greater than (or equal to) the minimum O-D generalized travel time. This is satisfied with the first equilibrium principle of Wardrop.

Similarly, with (

if

if

That is to say, when shelter

From (

if

if

The penalty coefficient of shelter

Similarly, from (

if

if

That is, the queuing delay on link

Equations (

By solving the lower programming model, the link flows and path flows (the result of path flows is not unique) that meet the above conditions can be obtained. But if the demand is too large and exceeds the capacity of the whole traffic network, the model will have no solution.

This section aims to design an algorithm to effectively solve the proposed bilevel model. Since bilevel programming is a NP-hard problem, which is difficult to solve by traditional optimization algorithms or analytical methods, the usual solving methods are based on heuristic algorithms. Therefore, in this paper, iterative optimization algorithm is adopted in the upper model. Besides, the addition of constraint conditions in the lower model changes the original Descartes form of UE model, which results in the fact that the lower model cannot be solved by traditional F-W method. Commonly used methods include penalty function method and Lagrange multiplier method. In this paper, the lower model is transformed and then solved by dynamic penalty function algorithm proposed by Zhang [

The specific steps of the iterative optimization algorithm are shown in Algorithm

Set

and mutation probability

Generate

represented by

For each lane distribution scheme:

(3.1) Calculate the number of lanes on each link

(3.2) Run the lower programming model according to demand

(3.3) If any solution exists, store the corresponding lane distribution scheme into the set of historical feasible schemes

Turn to Step

If all schemes can’t find a feasible solution in the lower level model, they should be renewed according to following procedure:

(4.1) Renew the update count of lane schemes,

(4.2) If there are more than

algorithm, then obtain

(4.3) If there are less than

to all of these schemes. Besides, generate

(4.4) Turn to Step

Let

The process of local searching algorithm in Algorithm

For each road section, judge the flow distribution results of a certain historical feasible scheme; if

As for the lower level model, the topology of the origin network is firstly transferred. Augment the original network with a dummy node, denoted

This simple modification of the network topology can transform the combined distribution/assignment problem with constraint of traffic capacity into an equivalent UE problem with constraint of traffic capacity. Based on the modification, a dynamic penalty function algorithm is used to solve the problem.

The basic idea of the dynamic penalty function algorithm is to transform equivalent UE problem with constraint into traditional UE problem by adding penalty function. In UE problem with constraint, the generalized travel time

Function

monotone, continuous, and derivable;

when

During the problem solving process, both

Due to the poor convergence performance of the traditional Frank-Wolfe method for solving UE model, the linear decomposition algorithm proposed by Aashitiani and Magnanti [

The specific steps of the dynamic penalty function algorithm for the lower level model are shown in Algorithm

(1.1) Set

convergence criteria

(1.2) Set valid paths of each O-D pair

(2.1) Calculate the generalized travel time

between each OD pair;

(2.2) Apply all-or-nothing assignment to get the initial path flow

(3.1) Update the set of valid paths

time

(3.2) Linearize the NCP problem composed of effective paths at

[

(3.3) If the convergence criterion Eq. (

If the convergence criterion Eq. (

In this section, the above proposed model and algorithm will be tested and evaluated in an experimental network. First the topology of network and relevant information is given, then the analyzation of the results.

Take the test network in the Figure

Information of link properties.

Link no. | Starting node | Ending node | Number of lanes | Free-flow travel time ( | Capacity |
---|---|---|---|---|---|

1 | 1 | 2 | 3 | 1.5 | 900 |

2 | 2 | 1 | 3 | 1 | 900 |

3 | 2 | 3 | 3 | 0.8 | 900 |

4 | 3 | 2 | 3 | 0.8 | 900 |

5 | 1 | 4 | 2 | 1 | 600 |

6 | 4 | 1 | 2 | 0.8 | 600 |

7 | 2 | 5 | 4 | 1.25 | 1200 |

8 | 5 | 2 | 4 | 1 | 1200 |

9 | 3 | 6 | 3 | 1.25 | 900 |

10 | 6 | 3 | 3 | 2 | 900 |

11 | 4 | 5 | 4 | 1.5 | 1200 |

12 | 5 | 4 | 4 | 1.25 | 1200 |

13 | 5 | 6 | 4 | 0.75 | 1200 |

14 | 6 | 5 | 4 | 0.6 | 1200 |

15 | 4 | 7 | 3 | 0.8 | 900 |

16 | 7 | 4 | 3 | 0.9 | 900 |

17 | 5 | 8 | 4 | 1.25 | 1200 |

18 | 8 | 5 | 4 | 1.2 | 1200 |

19 | 6 | 9 | 3 | 1.1 | 900 |

20 | 9 | 6 | 3 | 1 | 900 |

21 | 7 | 8 | 3 | 0.95 | 900 |

22 | 8 | 7 | 3 | 0.9 | 900 |

23 | 8 | 9 | 3 | 1.2 | 900 |

24 | 9 | 8 | 3 | 1.3 | 900 |

Information of evacuation demand.

No. | Origin node | Demand ( |
---|---|---|

1 | 1 | 1500 |

2 | 4 | 1200 |

Information of shelter capacity.

No. | Destination node | Capacity ( |
---|---|---|

1 | 3 | 1000 |

2 | 6 | 800 |

3 | 9 | 1200 |

Topology of tested network.

Modify the original network by adding a dummy node 10 and connect it with three shelters by three dummy links. The topology of modified network is shown in Figure

Topology of modified network.

Path details of OD pair 1-10 at equilibrium.

Path No. | Path flow | Composition of path | Travel time | Queuing delay | Generalized travel time |
---|---|---|---|---|---|

1 | 900 | 1,3,25 | 2.645 | 2.1993 | 4.8443 |

2 | 557.6044 | 5,15,21,23,27 | 4.2837 | 0.5606 | 4.8443 |

3 | 42.3956 | 5,11,13,26 | 3.7375 | 1.1068 | 4.8443 |

4 | 0 | 5,11,7,3,25 | 5.0450 | 1.0999 | 6.1449 |

5 | 0 | 1,7,13,26 | 3.8375 | 2.2060 | 6.0435 |

6 | 0 | 5,15,21,17,13,9,25 | 6.4106 | 0.5606 | 6.9712 |

Path details of OD pair 4-10 at equilibrium.

Path No. | Path flow | Composition of path | Travel time | Queuing delay | Generalized travel time |
---|---|---|---|---|---|

1 | 757.6044 | 11,13,26 | 2.5875 | 1.1065 | 3.6940 |

2 | 42.3956 | 15,21,23,27 | 3.1337 | 0.5603 | 3.6940 |

3 | 400 | 11,13,19,27 | 3.6939 | 0.0001 | 3.6940 |

4 | 0 | 11,13,10,25 | 4.5875 | 0 | 4.5875 |

5 | 0 | 6,1,3,25 | 3.4450 | 2.1993 | 5.6443 |

6 | 0 | 15,21,23,19,9,25 | 5.4901 | 0.5603 | 6.0504 |

Although the path flow results of traffic assignment in crowded traffic networks with queuing delays are not necessarily unique, the result satisfies that generalized travel time of all the selected paths is equal, which meets the conditions of UE equilibrium.

Detailed information for OD pairs.

No. | Origin node | Destination node | Demand | Minimum generalized travel time |
---|---|---|---|---|

1 | 1 | 3 | 900 | 4.8443 |

2 | 1 | 6 | 42.3956 | 4.8443 |

3 | 1 | 9 | 557.6044 | 4.8443 |

4 | 4 | 3 | 0 | 3.8375 |

5 | 4 | 6 | 757.6044 | 3.6940 |

6 | 4 | 9 | 442.3956 | 3.6940 |

That means for a certain origin, generalized travel time of all the selected shelters is equal, and generalized travel time of shelters which are not chosen is larger than the chosen shelters.

Table

Detail information for shelter choice.

No. | Destination node | Capacity | Demand | Queuing Delay |
---|---|---|---|---|

1 | 3 | 1000 | 900 | 0 |

2 | 6 | 800 | 800 | 1.1064 |

3 | 9 | 1200 | 1000 | 0 |

All the experimental results are consistent with the above analysis of first-order conditions of the Lagrangian function.

Information of different demand and shelter capacity scenarios.

Demand scenarios | Shelter capacity scenarios | ||||
---|---|---|---|---|---|

Origin node | Demand ( | Destination node | Capacity ( | ||

D1 | 1 | 1500 | S1 | 3 | 1000 |

4 | 1200 | 6 | 800 | ||

D2 | 1 | 700 | 9 | 1200 | |

4 | 2000 | S2 | 3 | 3000 | |

D3 | 4 | 700 | 6 | 800 | |

7 | 2000 | 9 | 3000 |

Information of combination of different scenarios.

No. | Demand scenario | Shelter capacity scenario |
---|---|---|

Case 1 | D1 | S1 |

Case 2 | D1 | S2 |

Case 3 | D2 | S2 |

Case 4 | D3 | S2 |

In the solving algorithm, set step size

Feasible lane distribution scheme and link flow in Case 1 when

Link No. | Number of lanes | Capacity | Link flow |
---|---|---|---|

1 | 5 | 1500 | 1065 |

2 | 1 | 300 | 0 |

3 | 4 | 1200 | 1000 |

4 | 2 | 600 | 0 |

5 | 2 | 600 | 600 |

6 | 2 | 600 | 0 |

7 | 6 | 1200 | 65 |

8 | 2 | 600 | 0 |

9 | 5 | 1500 | 0 |

10 | 1 | 300 | 0 |

11 | 5 | 1500 | 1332 |

12 | 3 | 900 | 0 |

13 | 7 | 2100 | 1100 |

14 | 1 | 300 | 0 |

15 | 2 | 600 | 600 |

16 | 4 | 1200 | 0 |

17 | 4 | 1200 | 297 |

18 | 4 | 1200 | 0 |

19 | 1 | 300 | 300 |

20 | 5 | 1500 | 0 |

21 | 3 | 900 | 600 |

22 | 3 | 900 | 0 |

23 | 3 | 900 | 897 |

24 | 3 | 900 | 0 |

Detailed information for shelter choice in Case 1 when

No. | Destination node | Capacity ( | Demand ( |
---|---|---|---|

1 | 3 | 1000 | 997 |

2 | 6 | 800 | 800 |

3 | 9 | 1200 | 1200 |

In case 2, by increasing the shelter capacity, the maximum multiplier

Feasible lane distribution scheme and link flow in Case 2 when

Link No. | Number of lanes | Capacity | Link flow |
---|---|---|---|

1 | 6 | 1800 | 1800 |

2 | 0 | 0 | 0 |

3 | 6 | 1800 | 1800 |

4 | 0 | 0 | 0 |

5 | 4 | 1200 | 1200 |

6 | 0 | 0 | 0 |

7 | 4 | 1200 | 0 |

8 | 4 | 1200 | 0 |

9 | 1 | 300 | 0 |

10 | 5 | 1500 | 0 |

11 | 7 | 2100 | 2100 |

12 | 1 | 300 | 0 |

13 | 8 | 2400 | 2100 |

14 | 0 | 0 | 0 |

15 | 5 | 1500 | 1500 |

16 | 1 | 300 | 0 |

17 | 0 | 0 | 0 |

18 | 8 | 2400 | 0 |

19 | 5 | 1500 | 1300 |

20 | 1 | 300 | 0 |

21 | 6 | 1800 | 1500 |

22 | 0 | 0 | 0 |

23 | 6 | 1800 | 1500 |

24 | 0 | 0 | 0 |

Detailed information for shelter choice in Case 2 when

No. | Destination node | Capacity ( | Demand ( |
---|---|---|---|

1 | 3 | 3000 | 1800 |

2 | 6 | 800 | 800 |

3 | 9 | 3000 | 2800 |

In conclusion, the emergency reliability of the entire road network can be limited by both shelter capacity and road capacity. Although improving road capacity is a good way to improve network emergency reliability, sometimes, increasing the shelter capacity, there would also be a significant increase in the network emergency reliability.

By analyzing the solution results of all the above cases and combining the enumeration method, the true optimal value of multiplier

Comparison between calculated optimal solution and true optimal solution.

Calculated optimal | Computation time | True optimal | |
---|---|---|---|

Case 1 | 1.11 | 125 | 1.111111 |

Case 2 | 2.00 | 1125 | 2 |

Case 3 | 2.22 | 1340 | 2.222222 |

Case 4 | 1.80 | 940 | 1.8 |

The computation time of the proposed algorithm is mainly consumed in solving the lower level joint UE distribution/assignment problem. Therefore, in this paper, the times of solving the lower assignment problem are used to evaluate the solving efficiency of the proposed algorithm. Each case runs 10 times, and the average times to solve the lower assignment problem are listed in Table

Reliability is an important feature of transportation network. In order to evaluate the reliability of road network under emergency evacuation, a bilevel programming model is proposed in this paper. The concept of reserve capacity is adopted, and the influence of reversible lane measures taken under emergency conditions, as well as the capacity limits of links and shelters, are considered. An iterative optimization method is proposed to solve the upper level model, and the lower level model is transformed and then solved by a dynamic penalty function algorithm. Finally, the numerical example demonstrates the rationality of the proposed model and feasibility of the proposed solution algorithms.

The proposed lower level model can effectively reflect the route choice and shelter choice behavior of evacuees under emergency condition. Evacuees may choose to evacuate to another shelter if the shelter with smaller travel time is full. Besides, when the shortest path has reached its maximum capacity, evacuees are forced to choose other subprime paths. The assignment result accords with UE equilibrium.

Although the reserve capacity of the network can be improved by reversible lane measures, but both link capacity and shelter capacity could be the restriction of emergency reliability of the entire road network. In addition, the efficiency of the proposed algorithms is also verified in the case study.

The model proposed here is a little complicated and many practical problems have been simplified. In further study, the proposed model can be not difficult to contain congestion and delays at intersections. Additionally, uncertainty factors should be considered, including demand uncertainty, disaster location uncertainty, link capacity uncertainty, and so on.

All the data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This paper is supported by the Research Projects of Natural Science Foundation of Guangdong Province under Grant no. 2015A030310341, the Research Projects of Natural Science Foundation of Guangdong Province under Grant no. 2018A030313119, and the Research Project of Shenzhen Technology University.