Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 7896926, 34 pages
https://doi.org/10.1155/2018/7896926

Research Article

An Improved Shuffled Frog Leaping Algorithm and Its Application in Dynamic Emergency Vehicle Dispatching

Xiaohong Duan, Tianyong Niu, and Qi Huang

School of Economics and Management, North China University of Technology, Beijing 100144, China

Correspondence should be addressed to Xiaohong Duan; 12116350@bjtu.edu.cn

Received 22 November 2017; Revised 20 January 2018; Accepted 28 January 2018; Published 28 March 2018

Academic Editor: Guillermo Cabrera-Guerrero

Copyright © 2018 Xiaohong Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The traditional method for solving the dynamic emergency vehicle dispatching problem can only get a local optimal strategy in each horizon. In order to obtain the dispatching strategy that can better respond to changes in road conditions during the whole dispatching process, the real-time and time-dependent link travel speeds are fused, and a time-dependent polygonal-shaped link travel speed function is set up to simulate the predictable changes in road conditions. Response times, accident severity, and accident time windows are taken as key factors to build an emergency vehicle dispatching model integrating dynamic emergency vehicle routing and selection. For the unpredictable changes in road conditions caused by accidents, the dispatching strategy is adjusted based on the real-time link travel speed. In order to solve the dynamic emergency vehicle dispatching model, an improved shuffled frog leaping algorithm (ISFLA) is proposed. The global search of the improved algorithm uses the probability model of estimation of distribution algorithm to avoid the partial optimal solution. Based on the Beijing expressway network, the efficacy of the model and the improved algorithm were tested from three aspects. The results have shown the following: Compared with SFLA, the optimization performance of ISFLA is getting better and better with the increase of the number of decision variables. When the possible emergency vehicle selection strategies are 8¹⁵, the objective function value of optimal selection strategies obtained by the base algorithm is 210.10% larger than that of ISFLA. The prediction error of the travel speed affects the accuracy of the initial emergency vehicle dispatching. The prediction error of ±10 can basically meet the requirements of the initial dispatching. The adjustment of emergency vehicle dispatching strategy can successfully bypassed road sections affected by accidents and shorten the response time.

1. Introduction

Urban Expressway can ease the traffic pressure on large cities and plays an important role in the urban traffic system. However, with the increasing amount of traffic, accidents often happen on expressways and cause great damage to people’s life and property. Rapid emergency rescue can effectively reduce accident loss, and emergency vehicle dispatching is the key to emergency rescue.

Research of emergency vehicle dispatching started with assumptions of static travel time or distance, and the original dispatching problem mainly included two basic issues. In the problem with one accident, we only need to choose the nearest emergency vehicle to rescue the accident. The core is the shortest paths of emergency vehicles in the road network [1]. In the problem with multiple accidents, choosing the nearest emergency vehicle need not be an optimal decision [2]. We also need to select suitable emergency vehicles for different accidents to minimize their response times. It is a combinatorial optimization problem. Therefore, models for solving combinatorial optimization, such as Hungarian method, direct cost model, and opportunity cost model, were used to solve the emergency vehicle dispatching [3]. Aiming at the random resource requirements of potential incidents, Ozbay et al. used probabilistic constraint to improve the opportunity cost model [4]. Emergency vehicle dispatching with multiple accidents is a complex problem relating to various factors. In addition to response time, factors such as fairness, cost, and loss were considered to set up multiobjective dispatching models [5, 6]. In order to solve these NP-hard problems, ant colony algorithm [7], particle swarm optimization [8], genetic algorithm [9], and other intelligent optimization algorithms had been widely applied.

Emergency vehicle dispatching problem considering the dynamic change of link weight (travel time, distance) started in the late 1990s. Taking the minimum response time as objective, Zografos et al. [10] integrated routing and dispatching module to set up an emergency response decision support system. Haghani et al. [11] built a simulation model of emergency vehicle dispatching. The model can assist decision makers to select suitable emergency vehicles and guide them to avoid congested areas. Dan et al. [12] divided dynamic dispatching problem into a series of static problems. Dispatching strategy was updated based on the time axis. A multiobjective model was established to solve these static problems. Yang et al. [13] set up an online dispatching and routing model for emergency vehicles. One day was divided into a number of intervals, and dispatching strategy was updated according to link travel time in each interval. Fu et al. [14] calculated the earliest response time using iteration method. A dynamic emergency resource dispatching system was designed. This research is essentially a static method for solving the dynamic problem. Dispatching strategies are continuously adjusted based on real-time traffic data at each decision-making instant. They suppose traffic data remain unchanged in the decision-making horizon. Emergency vehicles may be stopped up on the way to accidents because of changes in road conditions.

If link weight is thought to be a time-dependence function, this network is a kind of time-dependence network. Research on the time-dependence network still focuses on universal shortest path problem. On the assumption of discrete link weight based on travel time, Cooke and Halsey [15] proposed a model of the time-dependence network. Iteration method was used to solve the shortest path problem. Kaufman and Smith [16] proved that the time-dependence shortest path problem can be solved by polynomials only when the network satisfies the first-in-first-out (FIFO) property. Most of the research on the shortest path of time-dependence traffic network is based on FIFO property [17]. However, road network with discrete link travel time proposed by Cooke is not FIFO [16]. In order to solve this problem, Duan et al. [18] proposed a universal shuffled frog leaping algorithm for solving the shortest path of the non-FIFO and FIFO network. Ichoua et al. [19] replaced link travel time with link travel speed to build time-dependent function. Travel time calculated by travel speed function changes continuously and satisfies the FIFO property. Since link travel speed cannot be known ahead of the decision-making instant, Ichoua et al. divided a day into three horizons to distinguish between congestion and free flow. During each time horizon, link travel speed remains unchanged. If this travel speed function is used to solve emergency vehicle dispatching problem, the whole dispatching process may be at a certain time horizon with constant link travel speed. This travel speed function cannot satisfy the requirement of emergency vehicle dispatching. On the assumption that link travel speed decreases continuously with the entry instant, Yuan and Wang [20] proposed an emergency vehicle routing model taking the shortest travel time as objective. Zhou et al. [21] built a multiobjective optimization model to solve the multiperiod dynamic emergency resource scheduling problems. In order to solve the model, a multiobjective evolutionary algorithm was proposed. Zhou and Liu [22] designed a multiagent genetic algorithm to solve the multiperiod emergency resource scheduling problem considering the uncertainty of traffic. The experimental results show that the multiagent genetic algorithm precedes genetic algorithm for the problem.

According to the review of the literature, link travel speed function can reflect the dynamic changes of road conditions. However, it is difficult to model link travel speed function and solve the dynamic dispatching problem, and the dynamic problem is usually divided into a series of static problems. It only can get a local optimal strategy in each horizon. In order to get the overall optimal strategy, real-time data (reflecting the real-time road condition at decision-making instant) fuses with prediction data (reflecting the change of road condition in the whole dispatching process) to establish the link travel speed function. Meanwhile, considering the unpredictable changes in road conditions during the dispatching process, dispatching strategies are adjusted according to real-time travel speed. Multiple-incident and multiple-response (MIMR) emergency vehicle dispatching discussed in this paper is NP-hard problem with large scale variables. The metaheuristic algorithm has advantages in solving the NP-hard problem. Shuffled frog leaping algorithm (SFLA) is a relatively new heuristic algorithm. It was first proposed and applied in water distribution network designed by Eusuff et al. [23, 24]. This algorithm combines the advantages of particle swarm optimization (PSO) and shuffled complex evolution (SCE) algorithm, and it has been proved that the algorithm has good performance in convergence speed and solution precision [25, 26]. It was used to solve many real-word problems such as job shop scheduling and cloud computing resource allocation [27–29].

According to the above, in this paper, the polygonal time-dependent function based on real-time and prediction link travel speed is built to simulate real road conditions in expressway network. Integrating routing and selection of emergency vehicles, a dynamic dispatching model is built. The model takes time-dependent travel speed, response time, time window, and accident severity as key factors to get the optimal strategy. And the dispatching strategy is adjusted when the new accidents happen. An improved shuffled frog leaping algorithm (IFSLA) is put forward to solve the dynamic dispatching model. The algorithm uses the probabilistic model of the distribution estimation algorithm to generate new frog population. It can avoid a local optimum of shuffled frog leaping algorithm.

2. Problem Statement

Based on graph theory, the expressway network is abstracted as a time-dependent directed network model as shown in Figure 1. is the node set. It consists of hubs and interchanges on the expressway. is the total number of nodes. The link arc between adjacent nodes and is road section . The expressway network is a directed network, so . Emergency vehicles must run along the direction of the road section, and they can change the direction at the nodes. is the interested time horizon. is the link travel time function defined in time horizon . It represents a time for the emergency vehicle, leaving at an instant , traveling from node to . , is always defined. For , is defined as infinity.

Figure 1: Expressway network model.

(1) Link Travel Time Function. Taking as the minimum time interval, is divided into discrete time intervals, that are , , . is the initial dispatching decision-making instant. is the last instant, and make sure it is large enough for the emergency vehicle to arrive at the accident during the time period . is the link travel speed function. It represents the average speed of the road section at . Based on the link travel speed function of Ichoua et al. [19], it is assumed that travel speed in each time interval changes in the form of the polygonal line, and the polygonal-shaped travel speed function is shown in Figure 2. At the decision-making instant , the real-time link travel speed is known. However, the real-time link travel speed , , , , cannot be obtained. Therefore, they are approximated by the prediction travel speeds .

Figure 2: Polygonal-shaped travel speed function.

The polygonal-shaped travel speed function shown in Figure 2 can be expressed as

is continuous on , and it must be integrable on the interval . If the emergency vehicle enters the road section at the instant , its travel distance is a function of travel time .

in which,

Let formula (2) equal to the length of the road section ; the time when the emergency vehicle leaves the road section is

in which, is the inverse function of .

When the emergency vehicle enters the road section at instant , the travel time function of the road section is

(2) Dynamic Emergency Vehicle Dispatching Process. accidents are waiting for rescue at the instant , and they compose set . is the th accident, . The road section where the accident occurred is expressed as , and its location node is , its rescue time window is , and the required number of emergency vehicles is . represents the severity of the accident . is the emergency vehicle set, and , , stands for the th emergency vehicle. is the total number of emergency vehicles. The road section where the emergency vehicle is locat﻿ed is expressed as , and its location node is . When the emergency vehicle starts traveling at an instant , the shortest time path to the accident is , and the shortest travel time is .

The basic purpose of emergency vehicle dispatching is to shorten the response time of accidents. The response time mainly depends on the travel time of the emergency vehicles. In the complex expressway network, emergency vehicles can reach accidents through multiple paths. Therefore, the shortest time path problem should be solved and the shortest travel time from each emergency vehicle to each accident should be obtained firstly. Then, taking as the key factor, the suitable emergency vehicles are selected to rescue the accidents, that is, to solve the problem of emergency vehicle selection.

According to the emergency vehicle selection strategy, emergency vehicles head for accidents along the shortest paths. In this process, if there are no new accidents in the road network, travel speed function of each road section does not change obviously, but once an accident happened, travel speeds of road sections in the accident area will be greatly affected. In order to avoid the rescue delay caused by sudden changes in road conditions, it is necessary to update link travel speed function according to the real-time speed, and the dispatching decision should be adjusted. We only need to update the travel time functions of road sections where the new accidents are located and their upstream road sections. If the rescue paths of the accident go through the new accidents, the rescue strategy for the affected accident needs to be recalculated, and the rescue strategies for the new accidents need to be calculated too.

The dynamic dispatching process consists of two stages: At the decision instant , taking the polygonal-shaped travel speed function as the weight of the road section, the travel path from to is planned and the shortest travel time is obtained. Taking the shortest travel time as input, emergency vehicles are selected to rescue the accidents. At the instant , a new accident occurs in the road network. The dispatching strategy, including vehicle routing and selection, is dynamically adjusted according to the updated real-time link travel speed.

3. Dynamic Emergency Vehicle Dispatching Modeling

At the decision-making instant , the shortest travel time, the required number of emergency vehicles, the upper limit of rescue time window, and the accident severity are taken as the key factors. A dynamic dispatching model with vehicle routing is built. A list of all symbols is given in Symbols.

Formula (6) is the objective function of emergency vehicle dispatching. It consists of two parts, the total travel time for emergency vehicles to arrive at accidents and the punishment caused by the exceeding rescue time. In which, is a huge constant. At the initial decision-making instant , every accident needs to be rescued. At other decision-making instants, accident set includes new accidents that happened at and the accidents whose rescue paths at are affected by new accidents. Emergency vehicle set contains all emergency vehicles.

Formulas (7) are the emergency vehicle requirements constraints. They make sure that the vehicle requirements of each accident can be satisfied.

Formulas (8) are time window constraints of accidents. They guarantee that the latest arrival time of emergency vehicle does not exceed the upper limit of the time window .

Formulas (9) are constraints for the state of emergency vehicles. The emergency vehicle can only be dispatched to an accident or in an idle state.

Formula (10) is the constraint for the total number of emergency vehicles. It ensures that the total number of emergency vehicles dispatched to the accidents and in the idle state is .

Formulas (11) are constraints for the state of variables . At the decision-making instant , if the emergency vehicle , is dispatched to the accident , then ; otherwise, .

Formulas (12) are constraints for the state of variables . At the decision-making instant , if the latest rescue time for accident , exceeds the upper limit of rescue time window, then ; otherwise, .

Formulas (13) are constraints for the state of variables . At the decision-making instant , if the emergency vehicle , is in the idle state, then ; otherwise, .

Formula (14) is the objective function of rescue path for the accident . It minimizes the travel time of the emergency vehicle from node to the destination , where road sections and are connected.

Formula (15) calculates the instant when the emergency vehicle enters the road section .

Formulas (16)–(18) are connectivity constraints of the path. They ensure that there are no loops in the path sequence.

Formulas (19) ensure that the emergency vehicles do not pass road sections with accidents.

4. Solution for the Emergency Vehicle Dispatching Model

The shuffled frog leaping algorithm (SFLA) is a kind of metaheuristic algorithm that imitates the frog population’s behavior in obtaining food. The initial frog population is generated and divided into several memeplexes. Then the frogs search for the optimal solution within each memeplex for the defined number of times. Then frogs in different memeplex are shuffled so that the information can be exchanged globally. The group optimization and global information exchange alternate with each other until the convergence condition is satisfied. The mathematical model of SFLA is as follows.

(1) Initialization. frogs are randomly generated to compose the initial population . The position of the th frog is encoded as , , in which, is the dimension of the optimization problem. Each represents a feasible solution to the optimization problem. And each feasible solution corresponds to a performance function associated with the optimization objective.

(2) Rank and Grouping. frogs are ranked in descending order of performance function value. Position of the optimal frog in the population is marked. The population is divided into memeplexes, and there are frogs in each memeplex according to

(3) Local Search. Within each memeplex, the local optimization process is repeated for the specified number of iterations .

(3.1) Positions of the frogs in the memeplex, the best and the worst, are marked as and , respectively. is renewed along with according to

in which, is a random number, . is the adjustment of the th decision variable, . is the maximum adjustment of the th decision variable. is the renewed position of the th decision variable. and are the upper and lower limits of the position of the th decision variable.

(3.2) If the performance value of is better than , then ; otherwise, in formula (21) is replaced with , and the position updating is executed repeatedly.

(3.3) If the performance value of is still better than , then is substituted by a random frog location.

(4) Mix and Global Search. After a local search, all memeplexes are shuffled to form a new population. Frogs are ranked and the best frog is marked. Then the next grouping and local search are performed until the specified number of global iterations is completed.

4.1. Solution for the Emergency Vehicle Dispatching Model Based on Improved Shuffled Frog Leaping Algorithm

In order to solve the dynamic emergency vehicle dispatching model, an improved shuffled frog leaping algorithm (ISFLA) is proposed. As shown in Figure 3, the improved algorithm is guided by the idea of a base SFLA. The local search uses the optimization strategy of SFLA, and the global search uses the probability model of estimation of distribution algorithms (EDA) to avoid the partial optimal solution.

(a)

(b)

Figure 3: Principle of the ISFLA.

4.1.1. Encoding and Decoding

Decision variables of the emergency vehicle dispatching are 0-1 numerical variables , which express whether the vehicle should be dispatched to the accident . The working object of SFLA is integer vector, so it is necessary to convert to integer variable . represents the dispatching strategy of at the decision-making instant . Each vehicle can go to one of the accidents or in the idle state. The feasible set of is . The emergency vehicle is an idle vehicle while . In this case, the encoding of frog position can be expressed as a row vector matrix.

Similarly, should be decoded in the reverse way of encoding, and we can get the emergency vehicle dispatching strategy.

4.1.2. Performance Function

According to the objective function of the dynamic dispatching model of emergency vehicles, the performance function is defined as

Using the quantification method in the literature [30], the accident severity is divided into four levels, and the value is shown in Table 1.

Table 1: Value of accident severity.

Accident level
Description of the accident
Accident severity

Death toll
The number of serious injuries (SI)
The number of slight injuries
Property loss (L)

Minor accident
Situation 1
-
-
1-2
-
40

Situation 2
-
-
-
Motor vehicle accidents: <1,000
Nonmotor vehicle accidents: <200

Ordinary accident
Situation 1
-
1-2
-
-
60

Situation 2
-
-
3
-

Situation 3
-
-
-
<30,000

Major accident
Situation 1
1-2
-
-
-
80

Situation 2
-
3 ⩽ SI < 10
-
-

Situation 3
-
-
-
30,000 ⩽ < 60,000

Extra serious accident
Situation 1
3
-
-
-
100

Situation 2
-
11
-
-

Situation 3
1
8
-
-

Situation 4
2
5
-
-

Situation 5
-
-
-
60,000

4.1.3. Process of the Improved Shuffled Frog Leaping Algorithm

Process of the ISFLA is shown in Figure 4, and the specific process is as follows.

Figure 4: Process of the improved SFLA.

(1) Build Probability Model. The decision vector of emergency vehicle dispatching is , . dimensional matrix is used to describe the probability distribution of the frog population. is the element of the matrix . It represents the probability that the th decision variable is valued as .

(2) Generate the Initial Population. The value of the element is set as . According to a uniform distribution, frogs are generated randomly to form the initial population . Each frog’s position represents a feasible decision vector.

(3) Local Search. frogs are ranked in descending order of performance function value . Location of the optimal frog in the population is marked. The population is divided into memeplexes, and there are frogs in each memeplex according to formula (20). Within each memeplex, the local optimization process is repeated for the specified number of iterations according to formula (21)–(23).

(4) Select the Superior Individual and Update the Probability Model. The performance function value of each frog in the population is calculated. frogs are reordered in descending order of performance function value. The first frogs are selected to calculate the distribution of decision variables, and then the probability matrix is obtained. The updated probability model is

in which, is the forgetting factor.

(5) Generate New Population. According to the updated probability model , frogs are generated to form a new population .

5. Illustrative Examples

Beijing expressway network shown as in Figure 1 was taken as an example to demonstrate the efficacy of the model and the improved algorithm. The interested time horizon was 9:30 to 10:30 on December 1, 2016. The minimum time interval was 5 minutes. Real link travel speeds at the instant are given in Table 2. The upper limit of the time window was 50 minutes. Firstly, based on real link travel speeds at the instant , five examples with different scale variable were used to test the performance of the emergency vehicle dispatching based on ISFLA. Secondly, based on prediction link travel speeds with different error ranges, two examples were used to test the performance of the dynamic emergency vehicle dispatching based on the polygonal-shaped travel speed function. Thirdly, we simulated a scene of a new accident to demonstrate the efficacy of adjustment mechanism of dispatching strategy.

Table 2: The table gives travel speeds of at on December 1, 2016. In which, is the road section of Beijing expressway network. They are the basic data of illustrative examples in Section 5.

Road section
9:30
9:35
9:40
9:45
9:50
9:55
10:00
10:05
10:10
10:15
10:20
10:25
10:30

35.8215
20.3765
28.2279
31.8677
35.7043
38.8095
28.6015
17.5589
22.8039
20.5689
33.2365
23.6908
26.1908

22.5825
33.7016
14.5643
26.573
27.6807
19.0503
22.7254
23.13
22.2
21.6732
11.8164
23.3327
26.3327

19.7129
20.9999
15.7227
20.0905
17.6481
23.1212
26.4123
24.617
21.1975
17.4778
29.8401
32.2002
27.2002

15.355
15.528
11.5561
20.1183
19.7576
17.6415
22.0847
14.5041
14.1869
15.4142
16.9405
20.3074
22.5378

23.1756
27.1385
29.4663
30.5464
19.6817
24.3868
17.9294
18.4325
23.9658
31.6632
20.1517
25.8569
24.6569

26.4831
27.1754
21.7814
24.0784
17.1708
23.5244
17.2602
26.6113
35.9275
19.6655
23.9367
22.5875
17.5875

16.693
13.9778
11.2268
18.5879
15.5855
17.1857
13.8282
10.7979
18.6863
17.7514
22.7961
21.2068
16.2068

25.0379
30.4431
22.1758
18.9887
28.441
25.1131
27.0197
33.0966
27.4579
26.9891
33.9385
21.9777
26.9777

21.4607
30.1089
16.8812
11.7818
11.621
26.9056
21.581
39.804
35.43943
28.224
21.4316
34.3545
29.3545

23.7186
13.5768
22.6461
28.554
22.1646
13.8365
19.1874
18.0001
21.5321
21.6481
21.622
18.5449
23.5449

15.4424
20.2254
30.8515
25.626
23.4888
22.7928
25.596
25.40945
41.652
34.4304
25.5643
19.6756
22.6436

30.2678
19.3306
39.0199
32.8014
27.6075
30.1831
27.904
39.059
11.088
15.9983
27.7148
25.7315
28.6734

20.139
21.9476
18.3451
21.4544
12.4372
24.3021
13.8022
9.6058
25.6725
18.4364
20.9946
16.9337
11.9337

9.3128
23.4184
15.3372
48.2641
31.6018
36.0318
16.5346
22.9536
28.1376
21.2511
22.8202
18.4113
23.4113

35.7092
42.1002
34.1108
37.9314
35.7767
28.5644
32.7456
25.1776
36.9073
22.659
23.2064
25.022
27.0654

18.1438
22.07
25.7119
21.4686
15.3884
22.5308
15.1383
16.1952
17.7313
14.853
19.6452
16.5002
11.5002

15.6426
36.723
22.8957
30.8007
25.6438
32.4282
25.1974
49.986
33.9197
35.9149
21.4228
37.6876
32.6876

25.1075
31.5886
11.1249
26.73
24.013
15.8807
20.2512
24.1323
40.3112
36.2664
15.6697
26.0462
21.0462

24.2849
18.1288
17.2682
25.7382
30.8254
14.5333
20.753
22.5785
13.04
27.2062
20.9267
32.5609
27.5609

33.1513
25.6162
28.3083
24.1654
28.8115
17.0718
25.6522
24.2943
26.9422
30.1469
29.6986
38.7776
33.7776

45.289
42.7567
39.8016
45.4383
43.9181
54.312
36.4771
36.7876
46.5368
53.676
29.1054
41.8043
36.8043

24.2919
26.4158
27.8619
15.4707
24.595
9.864
24.684
7.38
22.212
32.076
21.8051
19.9445
23.3452

45.2188
42.6165
48.6611
48.0486
34.7927
42.0564
32.445
55.548
40.0004
29.4257
47.3986
39.5606
34.5606

24.8046
34.816
22.4496
28.5366
24.6233
16.668
12.8682
7.38
19.1072
28.6751
24.6707
18.2876
23.2876

23.6014
17.8158
18.7145
22.1506
22.0417
18.6709
23.9033
29.3326
21.7442
21.9771
11.9924
19.1274
17.1434

25.5525
22.7526
26.5187
19.5367
34.5426
25.6112
30.2352
23.8654
25.1875
29.4503
28.7788
20.3987
15.3987

31.4897
36.8907
32.8745
42.5121
33.336
34.4962
42.436
38.276
31.005
29.628
26.215
31.2124
26.2124

12.4945
15.9981
11.0279
18.8998
12.4372
25.544
14.796
22.2355
10.6805
15.7968
12.1055
28.9159
23.9159

26.3711
18.6249
20.6505
13.2397
22.2702
19.9457
12.22
18.1987
15.747
18.5462
16.7472
17.0505
12.0505

23.482
19.0404
27.7911
20.6294
22.1208
27.5455
18.4591
16.173
30.329
29.8818
23.7257
29.2399
34.2399

19.6502
21.1314
19.5799
16.8456
17.5475
29.628
21.7406
12.8068
26.6456
22.2932
27.195
24.6179
22.3426

11.6789
25.0173
23.1734
15.8519
10.4692
15.7769
17.2203
19.3803
17.6942
21.2171
20.9595
12.7466
17.7466

16.6906
23.4181
21.1797
23.4837
21.5237
16.9955
16.4308
22.9818
15.644
24.9333
22.9535
20.4386
15.4386

21.1883
19.3399
21.2697
30.4105
18.6214
20.5809
19.44
17.1289
23.0388
22.6296
31.6504
22.7549
27.7549

29.7455
23.1488
36.2439
41.5484
32.6593
48.132
30.6737
32.5309
32.5309
26.2273
31.0559
33.5541
28.5541

19.6753
16.1579
22.8854
19.7345
28.2388
23.4119
33.845
32.5529
25.8201
23.5862
30.7608
19.2675
23.2453

31.2132
24.0045
23.2547
21.9237
29.9919
39.1563
30.9103
24.9077
32.3465
24.4867
23.3021
31.8698
29.4568

33.088
35.2858
30.6669
28.0342
25.4681
38.1852
29.059
32.6558
38.8695
32.2736
27.1765
33.8246
28.8246

15.7673
18.1255
24.3937
18.8885
22.8062
14.7927
21.5846
19.7057
16.6918
25.0319
24.1279
23.1339
25.3462

15.5967
22.1842
30.6917
11.3833
18.6417
10.7881
24.5125
25.0496
9.6873
27.7184
12.5765
19.1341
14.1341

22.9506
21.4434
25.3072
15.7694
23.0083
22.2823
20.4405
28.2635
21.3286
27.7032
29.8792
24.2156
29.2156

21.9029
30.2519
25.8323
13.7023
33.4922
27.0706
25.3517
29.1525
26.6141
14.5851
11.252
22.3766
17.3766

11.1064
25.6402
15.2103
19.3209
26.4223
30.3127
13.0589
32.0853
27.2811
25.506
21.4571
20.4382
15.4382

15.4208
17.5974
24.8324
25.0641
21.4277
24.1301
23.3536
17.0912
21.1445
17.5043
14.5007
22.1737
27.1737

23.6147
38.1495
28.9343
24.2838
40.8111
24.048
28.2018
45.36
25.2878
27.65
23.2166
21.039
26.039

25.379
22.6816
16.5716
19.1937
21.2507
25.92
30.8738
16.8351
26.708
23.8517
20.5968
15.6582
20.6582

21.3262
21.9084
24.3039
28.8116
24.2594
30.2419
11.8582
19.7537
23.18
17.8241
15.6202
25.0852
20.0852

21.8971
11.6351
30.1314
25.5666
26.0133
29.028
19.9033
23.3528
32.7366
29.9893
20.5936
15.9837
21.2351

22.5953
26.6376
28.3237
25.4259
21.0793
19.422
15.9731
25.92
26.6161
46.296
29.4048
34.293
36.3256

24.8356
29.7005
35.0005
13.6867
36.468
22.2797
20.34
24.7755
11.088
33.676
30.3817
27.5295
32.5295

23.5872
31.4391
20.8336
9.2175
28.447
26.4114
17.0868
21.1131
23.8973
21.9235
23.4818
19.4364
14.4364

28.7868
28.331
27.7535
22.1266
29.8588
36.296
19.422
31.4111
25.0175
26.1084
44.1895
19.9332
25.3457

23.5312
20.7604
22.3994
35.63
34.2775
19.7144
21.6727
27.5377
26.1173
20.34
25.3507
30.9571
35.9571

26.7026
16.7753
19.1615
25.0103
22.2606
30.4504
30.7058
21.096
28.3757
30.9772
25.6049
32.1169
27.1169

16.9842
34.7854
32.1204
28.8798
29.7914
26.69
25.211
36.1121
26.7128
39.1083
29.6514
27.4658
22.4658

26.443
28.7525
32.2085
27.9107
33.5474
42.424
33.1352
44.132
29.628
21.285
38.88
22.4163
27.4163

20.0635
18.4469
18.9766
12.6472
14.5116
16.4666
13.5609
18.6594
28.8261
14.5888
20.2372
11.8279
6.8279

28.5716
27.6879
36.1665
27.9511
27.4823
33.0515
30.2954
22.6653
21.871
20.1762
43.026
34.8619
38.3462

17.1967
25.6923
22.2066
30.8889
40.716
15.114
12.96
5.544
25.92
13.0093
18.7247
23.7387
28.7387

32.0776
31.2593
49.3007
31.5098
56.466
46.3468
24.048
24.048
41.652
23.13
23.3579
31.7037
36.7037

21.765
13.6177
24.1532
20.6376
22.2606
17.6057
20.226
26.3626
7.6245
10.8335
21.0773
15.7262
17.5275

30.0455
34.0459
26.2879
28.8757
26.1564
33.9629
38.5092
29.0318
34.1819
29.4782
32.5105
37.9813
36.3864

9.971
19.224
11.088
24.048
14.796
24.048
21.3077
28.6925
20.7057
10.2956
16.9843
15.5002
10.5002

37.9415
31.3179
23.2697
28.7727
26.4532
38.88
27.6162
26.532
46.296
32.088
34.3308
27.6593
22.6593

24.5436
17.0723
22.9667
30.4918
19.9406
17.1968
15.9998
23.2988
27.0237
31.7541
19.3243
19.9782
20.2351

16.0336
21.0688
25.6279
30.6163
31.9925
22.4156
21.399
27.0876
25.4508
26.8263
26.8914
15.5985
20.5985

26.5774
22.212
17.5736
27.2318
15.442
13.6652
26.5167
17.5446
24.1079
30.6512
24.6187
27.79
32.79

31.4044
31.4625
47.808
37.008
40.1212
35.172
42.588
23.882
33.7562
36.6223
31.0834
29.3702
24.3702

19.8354
23.3835
28.8326
33.7234
19.1974
36.8871
32.6304
25.8425
23.494
21.5067
29.1639
21.527
26.527

28.3482
29.4714
19.5844
21.8987
25.765
28.5483
33.6437
26.5502
35.3078
31.3771
34.8706
29.6347
32.5326

26.0216
27.5469
21.8179
32.4478
33.4188
21.0689
23.4394
28.1423
24.1956
32.7357
29.0167
20.4584
29.5815

22.5705
28.1672
18.9582
31.5342
35.7092
23.1775
24.9478
28.0398
32.2195
28.1321
27.3823
22.1031
32.4264

32.0161
33.8281
49.6334
31.4378
24.217
41.5755
35.0492
28.7604
45.0867
34.7548
33.471
28.1108
23.1108

21.5587
24.7503
13.8207
28.1799
36.1223
28.797
22.4211
26.1431
19.8201
34.3893
29.7035
28.9367
33.9367

12.8279
35.172
20.358
10.17
23.1605
21.7582
22.824
29.3252
17.6377
28.4508
16.513
25.1189
30.1189

16.8102
19.7036
20.046
17.7061
18.0843
19.2021
21.7497
26.6898
22.7205
17.28
22.5881
17.5268
22.5268

26.3685
19.6544
20.9984
19.641
25.8781
19.1095
17.6822
25.7649
20.8829
24.8079
21.2406
24.3477
29.3477

16.8402
20.5886
16.65
29.8227
21.1968
19.2456
17.586
16.668
30.4934
20.9667
24.2786
25.0395
20.0395

29.2047
24.7494
18.3338
26.4002
14.7043
28.692
24.4668
11.088
20.34
25.92
27.8506
24.2783
19.2783

23.8144
22.3499
18.5552
20.8287
21.6968
11.4653
20.4559
26.4973
18.5754
24.9654
27.9486
28.4628
33.4628

30.4946
25.7359
36.4783
38.7628
25.6916
15.4647
23.175
20.7594
21.1762
19.3615
25.3909
35.5331
30.5331

36.1943
31.331
37.8856
28.1583
27.6649
37.044
39.9093
25.92
24.432
27.98
34.4391
29.0487
24.0487

27.5556
25.9729
24.751
29.2513
28.6489
25.1988
20.5482
25.7422
30.9605
16.9408
24.8777
15.2558
20.2558

26.3805
23.4738
24.2851
21.887
18.8264
21.3668
22.3002
12.6113
22.0732
26.5655
20.3389
30.4344
35.4344

25.331
28.2618
26.6353
30.4132
31.0304
22.7455
27.2853
21.2781
20.3911
31.6241
27.0002
26.7254
21.7254

16.8309
27.0856
23.9089
19.3628
24.066
34.794
20.89029
46.908
29.004
19.89
25.92
24.7748
29.7748

19.7948
24.4553
22.3961
13.2497
13.9135
29.1571
18.4512
35.4701
9.6681
25.6926
23.0038
18.4626
13.4626

27.6015
29.85
28.5529
18.1117
23.5015
44.424
33.3481
23.598
18.504
34.254
31.464
25.7385
20.7385

14.2224
19.9524
16.5509
16.6665
31.572
24.3207
20.976
20.5671
7.38
12.96
21.9918
24.4198
19.4198

33.5522
19.6369
25.2266
22.8186
25.8973
13.3329
34.1434
28.2749
37.0596
33.5
25.0689
19.3657
19.2819

30.6965
22.8707
28.1807
28.9659
20.5081
25.7812
32.9863
25.5622
23.4569
27.2833
23.6245
27.2157
32.2157

29.3327
37.7083
34.5393
35.3089
27.3833
33.152
55.548
36.1578
25.9193
31.8867
18.8907
25.4497
25.3257

34.5428
33.1044
26.3141
38.216
36.9792
19.828
37.179
26.652
21.296
28.5949
40.3886
37.2857
42.2857

22.6554
18.3384
20.6647
19.5223
20.0416
21.7215
15.3789
14.2179
21.1639
30.3106
17.1591
12.9237
14.3215

24.7294
13.4349
23.6613
27.7224
22.9866
23.672
35.172
25.92
15.732
7.38
23.5587
20.7311
25.7311

27.7343
45.2357
46.278
48.204
18.504
29.628
30.9123
31.895
42.7445
24.5717
19.9995
39.019
44.019

32.757
35.1901
29.7974
36.8838
39.0422
31.2206
43.6835
27.4834
34.6492
28.0919
34.9322
29.0565
24.0565

18.7413
22.9066
30.6725
31.2796
26.8011
27.437
29.1645
18.7048
23.0915
16.8493
23.7545
30.7464
32.7321

16.5841
23.1357
17.9721
5.544
24.984
26.1818
31.1185
28.4255
21.7963
16.2167
34.7601
26.723
21.723

35.7481
41.2172
50.004
58.632
23.6078
36.4782
55.548
46.296
34.4414
36.9133
41.7204
39.5694
44.5694

24.4832
35.7813
28.6492
24.5854
50.2455
34.5999
38.4233
43.1548
25.8914
38.4789
39.0105
36.4984
31.4984

32.8059
33.4404
29.0517
40.1148
33.0317
36.4324
28.7253
27.6238
32.5452
31.7361
38.9741
36.0303
41.0303

33.3402
32.9426
26.4765
40.716
32.813
22.212
29.628
27.7627
32.9475
25.4733
23.6324
29.8145
24.8145

18.0093
23.7247
25.822
28.7387
24.6348
33.8815
23.5953
27.6376
30.3237
27.4259
31.4048
28.6161
23.6161

29.9271
28.0907
21.013
30.969
20.7049
37.5447
31.1589
29.5889
25.6384
14.9421
20.8261
30.192
35.192

27.3122
23.0705
26.384
22.972
19.9255
23.7923
22.1811
20.1768
30.8551
23.1064
35.6519
20.5192
25.5192

13.3395
29.7268
20.8545
17.9808
23.7303
17.8568
24.8957
22.1967
36.8889
31.6923
13.5156
28.2066
26.3212

29.9233
27.5431
24.8307
20.7151
25.5793
26.5665
19.653
27.9609
23.8885
27.5288
29.6811
23.4767
28.4767

18.2443
21.0437
26.6463
22.7714
18.1332
14.5623
21.7784
22.6731
20.7887
22.4282
17.4275
27.2954
22.2954

34.6978
23.1543
29.0436
24.907
30.5601
28.0568
27.7801
42.7455
24.8498
27.7656
41.4151
22.2665
23.066

20.0793
14.9731
33.293
20.9773
25.5168
27.8158
25.568
20.4764
23.6614
25.3756
33.3948
21.6959
21.3256

22.4128
20.5054
20.34
35.796
14.796
45.548
29.3574
26.5325
26.7919
20.8342
33.6774
31.6365
26.6365

20.3312
24.4924
24.2836
31.2118
30.8979
28.6509
25.9583
22.4449
28.0319
34.8349
30.5642
29.0862
24.0862

34.5595
27.3606
36.8804
38.7388
41.0041
30.1619
38.1456
35.7044
34.9696
43.0544
21.1344
32.4679
27.4679

28.0225
19.644
36.4511
20.716
38.88
20.34
13.672
25.7334
31.2822
23.6385
24.025
29.1403
34.1403

16.0562
29.5635
23.2929
36.0355
39.4106
26.7698
36.1572
34.7219
20.5097
28.7999
30.5667
32.138
28.3215

28.3858
34.4957
41.3754
26.8214
20.2584
29.8203
17.0806
23.3569
26.6312
33.9373
33.6228
24.8346
19.8346

38.7342
42.1347
43.8738
47.8653
39.3315
48.2212
46.5461
55.2707
45.5078
46.0586
38.8005
37.3032
42.3032

45.4
34.8313
33.05829
51.8472
28.38
42.5736
49.98857
43.7484
47.011
41.9231
58.2984
34.9144
32.4512

28.2038
41.6719
31.5473
29.1606
45.1054
36.8593
36.2981
25.9666
34.6039
35.1209
25.4465
30.6452
25.6452

38.3401
35.6249
48.5614
38.4442
40.0985
32.3709
22.9813
29.4015
27.6955
29.0522
36.886
29.1026
24.1026

45.1644
46.6339
42.4113
35.9576
59.1703
28.4833
41.4302
35.7486
41.5471
48.225
46.9663
41.0249
46.0249

22.611
27.6803
34.0971
27.4104
16.7638
26.5984
23.4473
22.5357
22.6849
18.7214
22.0752
19.2935
21.2924

17.8407
29.7715
34.5095
28.9844
30.6518
27.3529
19.6213
29.6933
37.7258
43.1677
27.9886
19.6087
24.6087

43.7953
54.4549
61.5546
31.3509
68.5008
36.09
35.0472
66.11143
49.50514
41.346
35.172
57.6846
45.4344

19.3156
15.9166
19.7184
28.5772
26.2822
25.679
30.2439
26.0461
26.2169
33.6578
34.694
44.6195
49.6195

23.8205
32.8456
23.8767
28.0697
27.765
25.64229
23.42025
25.82571
30.67714
23.35015
32.391
32.8346
37.8346

34.1043
48.0765
36.7424
37.1687
44.4156
40.54
41.9454
47.7629
52.3747
42.4163
46.6214
42.0936
47.0936

32.6611
14.441
24.6137
33.4089
30.3681
28.0291
30.6215
34.7804
28.9226
23.4482
26.919
26.7533
31.7533

44.8318
51.211
52.254
60.2107
65.5371
60.4039
55.0154
64.5239
65.3911
59.9616
63.57
62.3024
67.3024

37.1429
49.9001
44.5609
36.0501
40.989
47.9961
38.0558
53.7996
41.8743
51.4709
38.3611
48.5935
45.4235

34.3629
28.4779
30.0763
29.3488
32.4467
35.5281
29.2485
33.7507
41.5317
36.5889
35.9516
26.1112
31.1112

47.4719
62.2148
68.7616
56.3855
66.4986
64.3643
51.5869
63.1797
57.5995
65.7155
60.4977
44.8243
49.8243

51.8907
47.8745
58.5121
49.4962
62.436
42.215
52.2124
45.918
53.8011
31.5118
53.7514
38.3837
42.3321

35.4259
28.2369
38.1587
33.4012
23.1637
18.4515
31.297
39.8748
34.087
33.3255
37.1265
24.6396
19.6396

48.7455
42.1488
55.2439
60.5484
52.6593
50.6737
52.5309
46.2273
52.0559
54.5541
56.3667
55.4897
50.4897

40.428
41.73709
51.84
36.22629
44.79943
56.88655
51.228
40.9608
48.7695
43.925
45.6818
52.9536
52.9536

21.5665
32.1018
33.315
38.2749
26.2941
36.7688
32.4286
27.3579
36.645
33.5939
30.5894
43.2638
38.2638

49.7784
60.5336
44.939
57.9558
64.3915
59.1353
62.1333
60.4039
47.4822
54.6687
67.7975
57.6513
59.6323

46.8069
54.2979
43.1809
42.617
57.6107
39.9223
43.9772
51.3589
46.1985
56.2395
53.273
47.3326
46.3216

23.5687
22.0959
23.0359
27.8531
24.22
21.4339
22.3374
17.0648
19.8429
16.522
21.9041
18.4894
17.4394

41.7558
50.9238
45.4815
42.6664
58.9132
52.9074
55.1869
59.0718
54.4421
57.4068
58.4637
62.3649
59.3321

22.452
37.3327
45.7083
42.5393
44.3089
36.3833
42.152
27.8907
35.9193
35.4497
46.1578
41.8867
36.8867

26.551
25.5558
22.89
17.6108
33.3904
42.0009
43.34
27.0469
29.724
27.2055
30.4557
24.4874
19.4874

50.6882
50.4938
70.0271
61.6079
51.544
60.1915
52.1194
60.6471
56.3798
69.9701
57.2431
46.103
51.103

51.9058
24.5301
43.5541
43.2413
65.1582
32.2113
45.666
53.676
42.588
29.616
47.6023
31.0086
36.0086

41.0666
47.6639
50.3153
57.3912
49.8147
42.576
56.672
57.4133
55.8927
59.5123
69.7826
64.2306
69.2306

28.6551
34.8394
33.4949
33.7306
45.8684
35.6184
47.1494
33.747
46.7805
34.822
40.4126
46.7659
41.7659

19.9468
15.8323
23.3384
13.9004
14.2217
23.352
29.1193
16.6816
15.9695
21.4484
25.4341
24.8079
29.8079

24.7969
26.8204
20.6999
26.4952
22.3768
23.7876
20.971
24.3051
47.4192
46.4452
32.8815
30.7235
25.7235

37.7425
56.6497
42.6488
44.8774
54.1986
44.5991
55.2349
44.2395
56.1024
53.0347
46.2959
41.0744
40.0421

33.3609
41.1878
35.4696
36.3263
38.1141
41.0098
49.226
44.7845
31.1314
26.2193
38.532
25.257
30.257

30.9449
30.5055
17.3013
27.1126
24.4442
14.7958
19.0287
25.0807
31.9526
25.6242
28.5735
21.6121
26.6121

35.3327
29.628
26.838
32.908
37.332
45.66
44.424
45.1164
37.9687
39.0396
37.2611
40.5312
35.5312

44.8368
46.52
38.0651
45.1987
48.8088
44.7385
57.8387
39.1315
39.1035
31.001
52.2839
51.9367
46.9367

11.9523
25.8677
29.0142
19.907
24.6575
26.7507
21.3123
24.9181
27.4727
19.3771
30.0109
31.5672
31.5672

30.0607
30.0464
30.0704
24.1245
28.2755
12.4826
18.9899
33.2118
22.0356
27.9654
35.0488
20.6665
22.6325

23.6209
18.1936
25.7786
21.5246
30.2617
13.5507
22.8474
29.7383
19.0304
18.2511
16.3982
20.3277
25.3277

5.1. Illustrative Examples of Emergency Vehicle Dispatching Based on the Improved Shuffled Frog Leaping Algorithm

(1) Calculating Link Travel Time Time-Dependent Function. Real link travel speeds at instant were used to build the polygonal-shaped travel speed function. Then the travel time function , , of each road section was calculated according to formula (5). Figure 5 shows travel speed time-dependent functions of road sections , , , , and Figure 6 shows corresponding travel time time-dependent functions of these road sections.

Figure 5: Link travel speed time-dependent functions.

Figure 6: Link travel time time-dependent functions.

(2) Parameters of Illustrative Examples of Emergency Vehicle Dispatching. We assume that there are 15 emergency vehicles at most in the expressway network. At 9:30, 7 accidents at most occurred simultaneously. Tables 3 and 4 list the parameters of the example.

Table 3: Accident parameters of the illustrative example.

Accident no.
Accident node
Road section of accident
Location of accident
Accident level
Accident severity
Accident demand

1

Minor
40
1

2

Ordinary
60
2

3

Major
80
2

4

Minor
40
1

5

Minor
40
2

6

Ordinary
60
2

7

Minor
40
1

Table 4: Emergency vehicle parameters of the illustrative example.

Vehicle no.
1
2
3
4
5

Vehicle node

Road section of vehicle

Location of vehicle

Vehicle no.
6
7
8
9
10

Vehicle node

Road section of vehicle

Location of vehicle

Vehicle no.
11
12
13
14
15

Vehicle node

Road section of vehicle

Location of vehicle

(3) Calculating the Shortest Travel Time for Each Emergency Vehicle to Each Accident. We assume 5 emergency vehicle dispatching examples with different complexity. The first example has 8 emergency vehicles and 3 accidents. Then, the numbers of emergency vehicles and accidents gradually increase, and the other four examples are formed. Emergency vehicles and accidents in each example are shown in Table 5.

Table 5: Emergency vehicles and accidents in each example.

Example no.
Vehicle no.
Accident no.

I
1, 2, 3, 4, 5, 6, 7, 8
1, 2, 3

II
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1, 2, 3, 4

III
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
1, 2, 3, 4, 5

IV
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
1, 2, 3, 4, 5, 6

V
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
1, 2, 3, 4, 5, 6, 7

The dynamic shortest path algorithm of emergency vehicles in expressway network proposed in our previous research [18] was used to calculate the shortest travel time and path, and the results are shown in Tables 6 and 7.

Table 6: The shortest travel time of the illustrative examples based on the real link travel speed.

Vehicle no.
Example no.
Accident no.

1
2
3
4
5
6
7

1
I
31.2925
38.9665
54.9495
--
--
--
--

II
31.2925
38.9665
54.9495
54.2331
--
--
--

III
31.2925
38.9665
54.9495
54.2331
34.8666
--
--

IV
31.2925
38.9665
54.9495
54.2331
34.8666
45.2505
--

V
31.2925
38.9665
54.9495
54.2331
34.8666
45.2505
∞

2
I
22.1482
31.9124
∞
--
--
--
--

II
22.1482
31.9124
∞
28.8721
--
--
--

III
26.9763
31.9124
∞
28.8721
15.2541
--
--

IV
26.9763
31.9124
∞
28.8721
15.2541
∞
--

V
26.9763
31.9124
∞
28.8721
15.2541
∞
38.6391

3
I
48.3611
47.2027
33.8637
--
--
--
--

II
48.3611
47.2027
33.8637
59.0018
--
--
--

III
48.3611
47.2027
33.8637
59.0018
54.0921
--
--

IV
48.3611
47.2027
33.8637
59.0018
54.0921
28.6649
--

V
48.3611
47.2027
33.8637
59.0018
54.0921
28.6649
∞

4
I
41.0760
16.3454
53.4742
--
--
--
--

II
41.0760
16.3454
∞
10.1881
--
--
--

III
41.0760
16.3454
∞
10.1881
35.6330
--
--

IV
41.0760
16.3454
∞
10.1881
35.6330
∞
--

V
41.0760
16.3454
∞
10.1881
35.6330
∞
19.4956

5
I
49.9989
25.1126
38.7917
--
--
--
--

II
49.9989
25.1126
38.7917
19.8843
--
--
--

III
49.9989
25.1126
38.7917
19.8843
43.5353
--
--

IV
49.9989
25.1126
38.7917
19.8843
43.5353
45.7355
--

V
49.9989
25.1126
38.7917
19.8843
43.5353
45.7355
23.4806

6
I
44.4116
∞
26.3079
--
--
--
--

II
44.4116
∞
26.3079
∞
--
--
--

III
44.4116
∞
26.3079
∞
54.4887
--
--

IV
44.4116
∞
26.3079
∞
54.4887
21.4419
--

V
44.4116
∞
26.3079
∞
54.4887
21.4419
∞

7
I
43.6867
23.4594
56.5536
--
--
--
--

II
43.6867
23.4594
56.5536
14.4705
--
--
--

III
43.6867
23.4594
56.5536
14.4705
41.8074
--
--

IV
43.6867
23.4594
56.5536
14.4705
41.8074
∞
--

V
43.6867
23.4594
56.5536
14.4705
41.8074
∞
12.3566

8
I
42.6146
21.7823
∞
--
--
--
--

II
42.6146
21.7823
∞
21.3312
--
--
--

III
42.6146
21.7823
∞
21.3312
40.7481
--
--

IV
42.6146
21.7823
∞
21.3312
40.7481
∞
--

V
42.6146
21.7823
∞
21.3312
40.7481
∞
20.0396

9
II
37.8754
33.6324
∞
30.5210
--
--
--

III
37.8754
33.6324
∞
30.5210
42.3188
--
--

IV
37.8754
33.6324
∞
30.5210
42.3188
∞
--

V
37.8754
33.6324
∞
30.5210
42.3188
∞
29.6284

10
II
42.3397
40.7709
28.2020
52.5731
--
--
--

III
42.3397
40.7709
28.2020
52.5731
47.7124
--
--

IV
42.3397
40.7709
28.2020
52.5731
47.7124
24.0304
--

V
42.3397
40.7709
28.2020
52.5731
47.7124
24.0304
∞

11
III
∞
27.4119
40.8674
21.9920
45.8012
--
--

IV
∞
27.4119
40.8674
21.9920
45.8012
48.8987
--

V
∞
27.4119
40.8674
21.9920
45.8012
48.8987
25.3691

12
III
25.3758
49.7617
48.7390
∞
33.5190
--
--

IV
25.3758
49.7617
48.7390
∞
33.5190
32.6388
--

V
25.3758
49.7617
48.7390
∞
33.5190
32.6388
∞

13
IV
∞
34.0944
21.9164
31.4456
54.1145
37.1084
--

V
∞
34.0944
21.9164
31.4456
54.1145
37.1084
33.8540

14
IV
10.1994
32.9815
∞
48.1107
13.8316
54.7018
--

V
10.1994
32.9815
∞
48.1107
13.8316
54.7018
∞

15
V
43.5208
22.7684
∞
22.3348
41.7501
∞
20.7021

Table 7: The shortest travel time path of the illustrative examples based on the real link travel speed.

Vehicle no.
Example no.
Accident no.

1
2
3
4
5
6
7

1
I
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
--

II
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
3, 2, 1, 6, 18, 17, 16

III
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
3, 2, 1, 6, 18, 17, 16
3, 2, 1, 9
--
--

IV
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
3, 2, 1, 6, 18, 17, 16
3, 2, 1, 9
3, 12, 23, 37, 38

V
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
3, 2, 1, 6, 18, 17, 16
3, 2, 1, 9
3, 12, 23, 37, 38
BL

2
I
8, 9, 20, 21
8, 7, 18, 6
BL
--

II
8, 9, 20, 21
8, 7, 18, 6
BL
8, 7, 18, 17, 16

III
8, 9, 10, 21
8, 7, 18, 6
BL
8, 7, 18, 17, 16
8, 9
--
--

IV
8, 9, 10, 21
8, 7, 18, 6
BL
8, 7, 18, 17, 16
8, 9
BL

V
8, 9, 10, 21
8, 7, 18, 6
BL
8, 7, 18, 17, 16
8, 9
BL
8, 7, 18, 17, 16, 30, 47

3
I
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
--

II
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16

III
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16
13, 12, 11, 10, 9
--
--

IV
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16
13, 12, 11, 10, 9
13, 24, 38
--

V
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16
13, 12, 11, 10, 9
13, 24, 38
BL

4
I
17, 18, 6, 1, 2, 10, 21
17, 18, 6
17, 16, 15, 14, 27
--

II
17, 18, 6, 1, 2, 10, 21
17, 18, 6
BL
17, 16

III
17, 18, 6, 1, 2, 10, 21
17, 18, 6
BL
17, 16
17, 18, 7, 8, 9
--
--

IV
17, 18, 6, 1, 2, 10, 21
17, 18, 6
BL
17, 16
17, 18, 7, 8, 9
BL
--

V
17, 18, 6, 1, 2, 10, 21
17, 18, 6
BL
17, 16
17, 18, 7, 8, 9
BL
17, 31, 47

5
I
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
--

II
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
29, 15, 16

III
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
29, 15, 16
29, 15, 5, 6, 1, 9
--
--

IV
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
29, 15, 16
29, 15, 5, 6, 1, 9
29, 43, 42, 41, 40, 39, 38
--

V
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
29, 15, 16
29, 15, 5, 6, 1, 9
29, 43, 42, 41, 40, 39, 38
29, 30, 47

6
I
25, 24, 23, 22, 21
BL
25, 26, 27
--

II
25, 24, 23, 22, 21
BL
25, 26, 27
BL

III
25, 24, 23, 22, 21
BL
25, 26, 27
BL
25, 24, 23, 12, 11, 10, 9
--
--

IV
25, 24, 23, 22, 21
BL
25, 26, 27
BL
25, 24, 23, 12, 11, 10, 9
25, 24, 38
--

V
25, 24, 23, 22, 21
BL
25, 26, 27
BL
25, 24, 23, 12, 11, 10, 9
25, 24, 38
BL

7
I
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
--

II
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
31, 30, 16

III
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
31, 30, 16
31, 17, 18, 6, 1, 9
--
--

IV
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
31, 30, 16
31, 17, 18, 6, 1, 9
BL
--

V
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
31, 30, 16
31, 17, 18, 6, 1, 9
BL
31, 47

8
I
32, 33, 19, 20, 21
32, 18, 6
BL
--

II
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16

III
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16
32, 18, 6, 1, 9
--
--

IV
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16
32, 18, 6, 1, 9
BL
--

V
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16
32, 18, 6, 1, 9
BL
32, 31, 47

9
II
33, 19, 20, 21
33, 32, 18, 6
BL
33, 32, 18, 17, 16

III
33, 19, 20, 21
33, 32, 18, 6
BL
33, 32, 18, 17, 16
33, 19, 8, 9
--
--

IV
33, 19, 20, 21
33, 32, 18, 6
BL
33, 32, 18, 17, 16
33, 19, 8, 9
BL
--

V
33, 19, 20, 21
33, 32, 18, 6
BL
33, 32, 18, 17, 16
33, 19, 8, 9
BL
33, 32, 31, 47

10
II
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16

III
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16
13, 12, 11, 10, 9
--
--

IV
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16
13, 12, 11, 10, 9
13, 24, 38
--

V
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16
13, 12, 11, 10, 9
13, 24, 38
BL

11
III
BL
29, 15, 5, 6
29, 28, 27
29, 15, 16
29, 15, 5, 6, 1, 9
--
--

IV
BL
29, 15, 5, 6
29, 28, 27
29, 15, 16
29, 15, 5, 6, 1, 9
29, 43, 42, 41, 40, 39, 38
--

V
BL
29, 15, 5, 6
29, 28, 27
29, 15, 16
29, 15, 5, 6, 1, 9
29, 43, 42, 41, 40, 39, 38
29, 30, 47

12
III
23, 22, 21
23, 12, 3, 2, 1, 6
23, 24, 25, 26, 27
BL
23, 12, 11, 10, 9
--
--

IV
23, 22, 21
23, 12, 3, 2, 1, 6
23, 24, 25, 26, 27
BL
23, 12, 11, 10, 9
23, 37, 38
--

V
23, 22, 21
23, 12, 3, 2, 1, 6
23, 24, 25, 26, 27
BL
23, 12, 11, 10, 9
23, 37, 38
BL

13
IV
BL
28, 29, 15, 5, 6
28, 27
28, 29, 15, 16
28, 29, 15, 5, 6, 1, 9
28, 42, 41, 40, 39, 38
--

V
BL
28, 29, 15, 5, 6
28, 27
28, 29, 15, 16
28, 29, 15, 5, 6, 1, 9
28, 42, 41, 40, 39, 38
28, 29, 30, 47

14
IV
10, 21
10, 2, 1, 6
BL
10, 9, 8, 7, 18, 17, 16
10, 9
10, 21, 35, 36, 37, 38
--

V
10, 21
10, 2, 1, 6
BL
10, 9, 8, 7, 18, 17, 16
10, 9
10, 21, 35, 36, 37, 38
BL

15
V
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16
32, 18, 6, 1, 9
BL
32, 31, 47

Note. BL means that the travel time of the optimal path exceeds the upper limit of the time window.

The expressway network is a directed network. When the emergency vehicle , goes to the accident , it must run along the direction of the road section to arrive at the node and then choose the next road section. In which, is the end of the road section where the emergency vehicle is located. Road section from the emergency vehicle’s location to node is a necessary way for emergency vehicle . In the same way, the emergency vehicle must arrive at the node and then run along the direction of the road section to reach the accident node . In which, is the start of the road section where the accident is located. Road section from the node to the accident’s location is a necessary way for emergency vehicle too. We only need to calculate the optimal path from the node to the node , so the optimal path from the node to the node is given in Table 7.

(4) Calculating the Optimal Dispatching Strategy. ISFLA was used to find the optimal dispatching strategy of the five examples. The improved algorithm was compared with SFLA. After some testing, parameters of ISFLA were defined as in Table 8. After ten runs, the best solutions of the five examples obtained by using these two algorithms are summarized in Table 9. Figure 7 shows the evolutionary processes of the two algorithms of example I and example V.

Table 8: Parameter selection of ISFLA.

Example no.

I
150
15
10
150
3
4
1
8
0.05

II
150
15
10
350
4
5
1
10
0.05

III
300
20
15
600
5
6
1
12
0.05

IV
374
22
17
800
5
7
1
14
0.05

V
500
25
20
900
6
8
1
15
0.05

Table 9: Optimal emergency vehicle dispatching strategy of the illustrative examples based on the real link travel speed.

Example no.
Example parameters
Algorithm
The optimal solution
Optimal emergency vehicle dispatching strategy
Travel time (min)

I
Total number of vehicles
8
ISFLA
0,1,3,2,0,3,0,2
to :
22.1482

to :
16.3454

Total number of accidents
3
to :
21.7823

to :
33.8637

Accident severity

to :
26.3079

SFLA
0,1,3,2,0,3,0,2
to :
22.1482

to :
16.3454

Accident demand

to :
21.7823

to :
33.8637

to :
26.3079

II
Total number of vehicles
10
ISFLA
0,1,0,2,0,3,4,2,0,3
to :
22.1482

to :
16.3454

Total number of accidents
4
to :
21.7823

to :
26.3079

Accident severity

to :
28.2020

to :
14.4705

SFLA
0,1,0,4,2,3,0,0,2,3
to :
22.1482

to :
25.1126

Accident demand

to :
33.6324

to :
26.3079

to :
28.2020

to :
10.1881

III
Total number of vehicles
12
ISFLA
5,5,0,2,0,3,4,2,0,3,0,1
to :
34.8666

to :
15.2541

to :
16.3454

Total number of accidents
5
to :
26.3079

to :
14.4705

to :
21.7823

Accident severity

to :
28.2020

to :
25.3758

SFLA
0,2,3,5,0,3,4,5,0,2,0,1
to :
31.9124

to :
33.8637

to :
35.6330

Accident demand

to :
26.3079

to :
14.4705

to :
40.7481

to :
40.7709

to :
25.3758

IV
Total number of vehicles
14
ISFLA
0,5,6,2,0,6,4,2,0,3,0,1,3,5
to :
15.2541

to :
28.6649

to :
16.3454

to :
21.4419

Total number of accidents
6
to :
14.4705

to :
21.7823

to :
28.2020

to :
25.3758

Accident severity

to :
21.9164

to :
13.8316

SFLA
6,0,0,5,0,6,2,2,1,5,4,3,3,0
to :
45.2505

to :
35.6330

to :
21.4419

to :
23.4594

Accident demand

to :
21.7823

to :
37.8754

to :
47.7124

to :
21.9920

to :
48.7390

to :
21.9164

V
Total number of vehicles
15
ISFLA
0,5,3,2,4,6,7,2,0,6,0,1,3,5,0
to :
15.2541

to :
33.8637

to :
16.3454

to :
19.8843

to :
21.4419

Total number of accidents
7
to :
12.3566

to :
21.7823

to :
24.0304

to :
25.3758

to :
21.9164

Accident severity

to :
13.8316

SFLA
1,2,0,7,4,3,7,5,2,6,3,0,6,5,5
to :
31.2925

to :
31.9124

to :
19.4956

to :
19.8843

to :
26.3079

to :
12.3566

Accident demand

to :
40.7481

to :
33.6324

to :
24.0304

to :
21.9920

to :
37.1084

to :
13.8316

to :
41.7501

Example no.
The total number of feasible solutions
Algorithm
Objective function value
The average evolution time (s)
Success rate (%) in ten runs

I
4⁸
ISFLA
7987
2.723
100

SFLA
7987
2.809
30

II
5¹⁰
ISFLA
8113
15.643
100

SFLA
9179
5.324
0

III
6¹²
ISFLA
10247
25.534
100

SFLA
13824
25.163
0

IV
7¹⁴
ISFLA
12061
97.333
100

SFLA
18097
51.567
0

V
8¹⁵
ISFLA
12947
144.199
100

SFLA
40149
101.786
0

Figure 7: Evolutionary processes of the two algorithms for solving the illustrative examples.

(5) Result Analysis. The applicability of the model and the improved algorithm for solving the emergency vehicle dispatching problems with different complexity were verified by the five illustrative examples. Under the constraints of emergency vehicle requirements and time windows, the optimal dispatching strategy takes accident severity as key factor to optimize the total travel times of emergency vehicles. The results correspond to the decision rules of emergency vehicle dispatching.

In example I, there are possible emergency vehicle selection strategies. Analyzing Table 9 shows that SFLA and ISFLA get the same optimal selection strategies. Their objective function values are 7987.

In example II, there are possible emergency vehicle selection strategies. Analyzing Table 9 shows that the objective function of optimal selection strategy obtained by using ISFLA is 8113. The SFLA is unwanted early convergence. Its objective function value of optimal selection strategies is 13.14% larger than that of ISFLA.

In example III, there are possible emergency vehicle selection strategies. Analyzing Table 9 shows that the objective function of optimal selection strategy obtained by using ISFLA is 10247. The SFLA is unwanted early convergence. Its objective function value of optimal selection strategies is 34.91% larger than that of ISFLA.

In example IV, there are possible emergency vehicle selection strategies. Analyzing Table 9 shows that the objective function of optimal selection strategy obtained by using ISFLA is 12061. The SFLA is unwanted early convergence. Its objective function value of optimal selection strategies is 50.04% larger than that of ISFLA.

In example V, there are possible emergency vehicle selection strategies. Analyzing Table 9 shows that the objective function of optimal selection strategy obtained by using ISFLA is 12947. The SFLA is unwanted early convergence. Its objective function value of optimal selection strategies is 210.10% larger than that of ISFLA.

Therefore, compared with SFLA, the optimization performance of ISFLA is getting better and better with the increase of the number of decision variables. ISFLA preceded SFLA in realizing the optimal resolution for the complicated emergency vehicle selection problem.

5.2. Illustrative Examples of Dynamic Emergency Vehicle Dispatching Based on the Prediction Link Travel Speed

Examples I and II in Section 5.1 were used to test the performance of dynamic emergency vehicle dispatching based on the prediction link travel speed. Polygonal-shaped travel speed functions , were built using prediction travel speeds with different absolute error intervals (±3 km/h, ±5 km/h, and ±10 km/h). The present prediction methods can control the maximum absolute error of the prediction travel speed within ±10 km/h [31, 32], so the maximum prediction error was assumed as ±10 km/h. The prediction travel speed was randomly generated within the interval , in which, is the real vehicle speed at the instant . In order to analyze the effect of the decrease of prediction error on the initial emergency vehicle dispatching strategy, the two prediction errors ±3 km/h and ±5 km/h were assumed. Then the travel time functions , of each road section were calculated according to formula (5). The dynamic shortest path algorithm of emergency vehicles in expressway network proposed in our previous research [18] was used to calculate the shortest travel time and path, and the results are shown in Tables 10 and 11.

Table 10: The shortest travel time of the illustrative examples based on the prediction link travel speed.

Vehicle no.
Example no.
Prediction error (km/h)
Accident no.

1
2
3
4

1
I
0
31.2925
38.9665
54.9495
--

±3
31.2737
38.5534
54.9460
--

±5
32.1368
38.4649
55.5341
--

±10
31.5763
42.7993
50.3388
--

II
0
31.2925
38.9665
54.9495
54.2331

±3
31.2737
38.5534
54.9460
53.9916

±5
32.1368
38.4649
55.5341
53.4761

±10
31.5763
42.7993
50.3388
∞

2
I
0
22.1482
31.9124
∞
--

±3
20.8670
32.3468
∞
--

±5
23.1840
31.3765
∞
--

±10
18.3767
32.2828
∞
--

II
0
22.1482
31.9124
∞
28.8721

±3
20.8670
32.3468
∞
28.8694

±5
23.1840
31.3765
∞
30.9670

±10
18.3767
32.2828
∞
26.4442

3
I
0
48.3611
47.2027
33.8637
--

±3
46.8840
47.3166
33.7495
--

±5
46.8704
47.2581
33.8035
--

±10
46.4381
52.2131
33.2153
--

II
0
48.3611
47.2027
33.8637
59.0018

±3
46.8840
47.3166
33.7495
59.8245

±5
46.8704
47.2581
33.8035
∞

±10
46.4381
52.2131
33.2153
58.7999

4
I
0
41.0760
16.3454
53.4742
--

±3
41.2953
16.4075
54.8336
--

±5
41.6761
16.6789
52.0917
--

±10
40.3378
17.3145
53.4260
--

II
0
41.0760
16.3454
∞
10.1881

±3
41.2953
16.4075
∞
10.5117

±5
41.6761
16.6789
∞
9.5027

±10
40.3378
17.3145
∞
9.6885

5
I
0
49.9989
25.1126
38.7917
--

±3
49.5269
25.4762
38.9094
--

±5
50.8201
24.7458
38.8310
--

±10
50.8567
28.2398
38.5163
--

II
0
49.9989
25.1126
38.7917
19.8843

±3
49.5269
25.4762
38.9094
19.9819

±5
50.8201
24.7458
38.8310
18.5017

±10
50.8567
28.2398
38.5163
22.5535

6
I
0
44.4116
∞
26.3079
--

±3
43.5208
51.2845
25.8029
--

±5
47.1051
51.4706
25.4729
--

±10
50.1697
54.9481
31.6198
--

II
0
44.4116
∞
26.3079
∞

±3
43.5208
51.2845
25.8029
59.7672

±5
47.1051
51.4706
25.4729
∞

±10
50.1697
54.9481
31.6198
∞

7
I
0
43.6867
23.4594
56.5536
--

±3
43.8486
23.8514
56.9816
--

±5
42.8287
22.6855
53.4435
--

±10
42.4865
24.9977
55.1983
--

II
0
43.6867
23.4594
56.5536
14.4705

±3
43.8486
23.8514
56.9816
14.7354

±5
42.8287
22.6855
53.4435
13.9424

±10
42.4865
24.9977
55.1983
13.5972

8
I
0
42.6146
21.7823
∞
--

±3
42.7782
21.8997
∞
--

±5
42.5819
21.8541
∞
--

±10
40.9011
26.2546
∞
--

II
0
42.6146
21.7823
∞
21.3312

±3
42.7782
21.8997
∞
21.5001

±5
42.5819
21.8541
∞
20.9878

±10
40.9011
26.2546
∞
22.7096

9
II
0
37.8754
33.6324
∞
30.5210

±3
37.3625
33.4413
∞
29.9943

±5
39.1213
34.4820
∞
31.7022

±10
37.5274
35.6110
∞
29.7247

10
II
0
42.3397
40.7709
28.2020
52.5731

±3
40.9388
40.6356
28.5534
53.1496

±5
42.4467
40.9227
28.6415
55.0761

±10
41.3142
41.4190
26.8230
51.2051

Table 11: The shortest travel time path of the illustrative examples based on the prediction link travel speed.

Vehicle no.
Example no.
Prediction error (km/h)
Accident no.

1
2
3
4

1
I
0
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
--

±3
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
--

±5
3, 12, 11, 22, 21
3, 2, 1, 6
3, 12, 13, 14, 27
--

±10
3, 12, 11, 22, 21
3, 2, 1, 6
3, 12, 13, 14, 27
--

II
0
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
3, 2, 1, 6, 18, 17, 16

±3
3, 2, 10, 21
3, 2, 1, 6
3, 12, 13, 14, 27
3, 2, 1, 6, 18, 17, 16

±5
3, 12, 11, 22, 21
3, 2, 1, 6
3, 12, 13, 14, 27
3, 2, 1, 6, 18, 17, 16

±10
3, 12, 11, 22, 21
3, 2, 1, 6
3, 12, 13, 14, 27
BL

2
I
0
8, 9, 20, 21
8, 7, 18, 6
BL
--

±3
8, 9, 20, 21
8, 7, 18, 6
BL
--

±5
8, 9, 20, 21
8, 7, 1, 6
BL
--

±10
8, 9, 20, 21
8, 7, 18, 6
BL
--

II
0
8, 9, 20, 21
8, 7, 18, 6
BL
8, 7, 18, 17, 16

±3
8, 9, 20, 21
8, 7, 18, 6
BL
8, 7, 18, 17, 16

±5
8, 9, 20, 21
8, 7, 1, 6
BL
8, 7, 18, 17, 16

±10
8, 9, 20, 21
8, 7, 18, 6
BL
8, 7, 18, 17, 16

3
I
0
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
--

±3
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
--

±5
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
--

±10
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
--

II
0
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16

±3
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16

±5
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
BL

±10
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 14, 15, 16

4
I
0
17, 18, 6, 1, 2, 10, 21
17, 18, 6
17, 16, 15, 14, 27
--

±3
17, 18, 6, 1, 2, 10, 21
17, 18, 6
17, 16, 15, 14, 27
--

±5
17, 18, 7, 8, 19, 20, 21
17, 18, 6
17, 16, 15, 14, 27
--

±10
17, 18, 6, 1, 2, 10, 21
17, 18, 6
17, 16, 15, 14, 27
--

II
0
17, 18, 6, 1, 2, 10, 21
17, 18, 6
BL
17, 16

±3
17, 18, 6, 1, 2, 10, 21
17, 18, 6
BL
17, 16

±5
17, 18, 7, 8, 19, 20, 21
17, 18, 6
BL
17, 16

±10
17, 18, 6, 1, 2, 10, 21
17, 18, 6
BL
17, 16

5
I
0
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
--

±3
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
--

±5
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
--

±10
29, 15, 5, 6, 1, 9, 20, 21
29, 15, 5, 6
29, 28, 27
--

II
0
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
29, 15, 16

±3
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
29, 15, 16

±5
29, 15, 5, 6, 1, 2, 10, 21
29, 15, 5, 6
29, 28, 27
29, 15, 16

±10
29, 15, 5, 6, 1, 9, 20, 21
29, 15, 5, 6
29, 28, 27
29, 30, 16

6
I
0
25, 24, 23, 22, 21
BL
25, 26, 27
--

±3
25, 24, 23, 22, 21
25, 24, 13, 4, 5, 6
25, 26, 27
--

±5
25, 24, 23, 22, 21
25, 24, 13, 4, 5, 6
25, 26, 27
--

±10
25, 24, 23, 22, 21
25, 24, 13, 4, 5, 6
25, 26, 27
--

II
0
25, 24, 23, 22, 21
BL
25, 26, 27
BL

±3
25, 24, 23, 22, 21
25, 24, 13, 4, 5, 6
25, 26, 27
25, 24, 13, 4, 5, 15, 16

±5
25, 24, 23, 22, 21
25, 24, 13, 4, 5, 6
25, 26, 27
BL

±10
25, 24, 23, 22, 21
25, 24, 13, 4, 5, 6
25, 26, 27
BL

7
I
0
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
--

±3
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
--

±5
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
--

±10
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 43, 42, 41, 27
--

II
0
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
31, 30, 16

±3
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
31, 30, 16

±5
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 28, 27
31, 30, 16

±10
31, 32, 33, 19, 20, 21
31, 17, 18, 6
31, 30, 29, 43, 42, 41, 27
31, 30, 16

8
I
0
32, 33, 19, 20, 21
32, 18, 6
BL
--

±3
32, 33, 19, 20, 21
32, 18, 6
BL
--

±5
32, 33, 19, 20, 21
32, 18, 6
BL
--

±10
32, 33, 19, 20, 21
32, 18, 6
BL
--

II
0
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16

±3
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16

±5
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16

±10
32, 33, 19, 20, 21
32, 18, 6
BL
32, 18, 17, 16

9
II
0
33, 19, 20, 21
33, 32, 18, 6
BL
33, 32, 18, 17, 16

±3
33, 19, 20, 21
33, 32, 18, 6
BL
33, 32, 18, 17, 16

±5
33, 19, 20, 21
33, 32, 18, 6
BL
33, 32, 31, 30, 16

±10
33, 19, 20, 21
33, 32, 18, 6
BL
33, 32, 18, 17, 16

10
II
0
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16

±3
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16

±5
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 4, 5, 15, 16

±10
13, 12, 11, 22, 21
13, 4, 5, 6
13, 14, 27
13, 14, 15, 16

Note. BL means that the travel time of the optimal path exceeds the upper limit of the time window.

ISFLA was used to find the optimal dispatching strategy of the two examples under three prediction errors, and the best solutions are shown in Table 12.

Table 12: Optimal emergency vehicle dispatching strategy of the illustrative examples based on the prediction link travel speed.

Example no.
Prediction error (km/h)
The optimal solution
Objective function value based on the prediction travel speed
Objective function value based on the real travel speed

I
0
0, 1, 3, 2, 0, 3, 0, 2
--
7987

±3
0, 1, 3, 2, 0, 3, 0, 2
7897
7987

±5
0, 1, 3, 2, 0, 3, 0, 2
7981
7987

±10
0, 1, 3, 2, 0, 3, 2, 0
8461
8088

II
0
0, 1, 0, 2, 0, 3, 4, 2, 0, 3
8113
8113

±3
0, 1, 0, 2, 0, 3, 4, 2, 0, 3
8071
8113

±5
0, 1, 0, 2, 0, 3, 4, 2, 0, 3
8126
8113

±10
0, 1, 0, 2, 0, 3, 4, 2, 0, 3
8569
8113

Example no.
Prediction error (km/h)
Optimal emergency vehicle dispatching strategy
Prediction travel time (min)
Real travel time (min)

I
0
to :
--
22.1482

to :
--
16.3454

to :
--
21.7823

to :
--
33.8637

to :
--
26.3079

±3
to :
20.8670
22.1482

to :
16.4075
16.3454

to :
21.8997
21.7823

to :
33.7495
33.8637

to :
25.8029
26.3079

±5
to :
23.1840
22.1482

to :
16.6789
16.3454

to :
21.8541
21.7823

to :
33.8035
33.8637

to :
25.4729
26.3079

±10
to :
18.3767
22.1482

to :
17.3145
16.3454

to :
24.9977
23.4594

to :
33.2153
33.8637

to :
31.6198
26.3079

II
0
to :
--
22.1482

to :
--
16.3454

to :
--
21.7823

to :
--
26.3079

to :
--
28.2020

to :
--
14.4705

±3
to :
20.8670
22.1482

to :
16.4075
16.3454

to :
21.8997
21.7823

to :
25.8029
26.3079

to :
28.5534
28.2020

to :
14.7354
14.4705

±5
to :
23.1840
22.1482

to :
16.6789
16.3454

to :
21.8541
21.7823

to :
25.4729
26.3079

to :
28.6415
28.2020

to :
13.9424
14.4705

±10
to :
18.3767
22.1482

to :
17.3145
16.3454

to :
26.2546
21.7823

to :
31.6198
26.3079

to :
26.8230
28.2020

to :
13.5972
14.4705

In example I, the optimal emergency vehicle dispatching strategy (including vehicle routing and selection), which is based on the prediction link travel speed with prediction errors of ±3, ±5, is the same as the actual optimal strategy. However, when the prediction error is ±10, the rescue vehicle for the accident is different from the actual optimal strategy. The emergency vehicle instead of the vehicle is dispatched to the accident . This led to a 23.4594 − 21.7823 = 1.6771 minutes extension of the rescue time for the accident . The objective function value based on the prediction travel speed is 101 more than that based on the real travel speed. It can be seen that the prediction error of the travel speed affects the accuracy of the initial emergency vehicle dispatching, but when the absolute prediction error is within ±10, the effect is not obvious.

In example II, the optimal emergency vehicle dispatching strategy (including vehicle routing and selection), which is based on the prediction link travel speed with prediction errors of ±3, ±5, and ±10, is the same as the actual optimal strategy.

It is known from the two examples that the prediction error of ±10 can basically meet the requirements of the initial dispatching.

5.3. Illustrative Example of Dynamic Adjustment of Dispatching Strategy

Example I in Section 5.2 was used to test the performance of dynamic adjustment of dispatching strategy. Suppose that the prediction error of link travel speed was ±5. At 9:30, emergency vehicles were dispatched to accidents according to the optimal dispatching strategy obtained in Section 5.2. Emergency vehicle was dispatched to accident . Emergency vehicle was dispatched to accident . Emergency vehicle was dispatched to accident . Emergency vehicle was dispatched to accident . Emergency vehicle was dispatched to accident . Emergency vehicles , , and were idle vehicles. The optimal dispatching strategy is shown in Figure 8.

Figure 8: Optimal emergency vehicle dispatching strategy at 9:30.

At 9:35, a new accident happens on the road section . According to the real link travel speed, location of each emergency vehicle was calculated. They are shown in Figure 9.

Figure 9: Locations of emergency vehicles and accidents at 9:35.

Due to the influence of the accident , road conditions of the accident section and its upstream sections and changed. In order to consider the impact of accident on upstream sections, a weakening factor to the travel speed of upstream section was introduced. Therefore, the travel speed functions of sections and were , , respectively. Suppose that decreases linearly with time . The initial value . When , .

The travel speed of each road section was recollected at 9:35, and the prediction travel speed function was calculated. We suppose that travel speed prediction error of road sections and was ±10 and that of other road sections was ±5. Then, travel paths of emergency vehicles were replanned, and the dispatching strategy was adjusted. The optimal dispatching strategy at 9:35 is shown in Figure 10. The dynamic emergency vehicle dispatching process is shown in Table 13.

Table 13: Dynamic emergency vehicle dispatching process.

Decision making instant
Vehicle no.
Road section of vehicle
Location of vehicle
The optimal solution
Optimal emergency vehicle dispatching strategy

9:30
1

0,1,3,2,0,3,0,2
to :  
to :  
to :  
to :  
to :

2

3

4

5

6

7

8

9:35
1

1,4,3,2,0,3,0,2
to :  
to :  
to :  
to :  
to :  
to :

2

3

4

5

6

7

8

Figure 10: Optimal emergency vehicle dispatching strategy at 9:35.

When the accident happened at 9:35, the dynamical adjustment mechanism timely responded to sudden changes in road conditions of the optimal path of emergency vehicle . After recalculating the dispatching strategy, the idle emergency vehicle replaced the emergency vehicle to rescue the accident . The emergency vehicle successfully bypassed road sections and to ensure the time limit for emergency rescue. Meanwhile, the emergency vehicle with the shortest travel time was dispatched to the new accident . Therefore, the adjustment of emergency vehicle dispatching strategy can effectively shorten the response time of accident.

6. Conclusions

The emergency vehicle dispatching in urban expressway network includes dynamic routing and emergency vehicles election. And the dispatching strategy needs to be adjusted according to the dynamic road conditions. Firstly, polygonal-shaped travel speed function based on real-time and prediction link travel speed is set up, and the dynamic emergency vehicle dispatching process is illustrated. Secondly, taking the accident severity as the key factor and the travel time as the objective function, a dynamic dispatching model considering the vehicle routing was established. The model consists of two stages: initial dispatching and dynamic adjustment. Thirdly, in order to avoid unwanted early convergence of SFLA in solving complex dispatching problems, the basic SFLA was combined with the probabilistic model of EDA, and the SFLA was improved. Finally, Beijing expressway network was taken as an example to test the efficacy of the model and the algorithm from three aspects. First, based on the real link travel speed, 5 emergency vehicle dispatching problems with different scale variable were modeled and solved by the improved shuffled frog leaping algorithm. Second, based on the prediction link travel speed and the real-time link travel speed, the dynamic emergency vehicle dispatching model and the improved SFLA were used to calculate the global optimal dispatching strategy. Third, a scene in which a new accident happened on the rescue path was simulated, and the dispatching strategy was adjusted. The results show the following: The emergency vehicle dispatching model can obtain the optimal dispatching strategy under the constraints of accident demands and time windows. For solving the complicated emergency vehicle selection problem, the improved SFLA has stronger search ability compared with the SFLA and can obtain more optimal dispatching strategy. The optimal strategy obtained by the emergency vehicle dispatching model based on the prediction link travel speed takes into account the dynamic changes in the road conditions during the whole dispatching process. The dynamic adjustment condition can timely respond to sudden changes in the road conditions and help emergency vehicles to avoid rescue delay.

Symbols

: | The expressway network node set
, : | The th expressway network node
: | The total number of nodes in the expressway network
: | The road section set of the expressway network
: | The road section form the node to the node
: | The interested time horizon
: | The link travel time function defined in time horizon . It represents a time for the emergency vehicle, leaving at an instant , from node to
: | The link travel time function set
: | The minimum time interval
: | The initial dispatching decision-making instant
, : | The start time of the th time interval
: | The last instant of the interested time horizon
: | The link travel speed function of the road section at the instant
: | The real-time link travel speed of the road section at the instant
: | The prediction travel speed at the instant calculated according to
: | The polygonal-shaped travel speed function at the instant
: | The length of the road section
: | The total number of accidents at the instant
: | The accident set at the instant
, : | The th accident at the instant
: | The road section where the accident occurred at the instant
: | The location node of the accident
: | The upper limit of rescue time window of the accident
: | The required number of emergency vehicles for the accident
: | The severity of the accident
: | The emergency vehicle set at the instant
, : | The th emergency vehicle at the instant
: | The total number of emergency vehicles at the instant
: | The road section where the emergency vehicle is located
: | The location node of the emergency vehicle
: | When the emergency vehicle starts traveling at an instant , the shortest time path to the accident
: | The travel time of the shortest time path at the instant
: | The th decision instant
: | The decision variable; at the decision-making instant , if the emergency vehicle , is dispatched to the accident , then ; otherwise,
: | The decision variable; at the decision-making instant , if the latest rescue time for accident exceeds the upper limit of rescue time window, then ; otherwise,
: | A huge constant
: | The accident set includes new accidents that happened at and the accidents whose rescue paths at are affected by new accidents
: | The latest rescue time of the accident
: | The decision variable; at the decision-making instant , if the emergency vehicle is in the idle state, then ; otherwise,
: | The link travel time function at the decision-making instant . It represents a time for the emergency vehicle, leaving at an instant , from node to
: | The total number of frogs
: | The frog population
= , , : | The position of the th frog
: | The dimension of the optimization problem
: | The performance function of the optimization problem
= , …: | The position of the optimal frog in the population
: | The total number of memeplexes
: | The total number of frogs in each memeplex
: | The total number of iterations within each memeplex
= , : | The position of the best frog in the memeplex
= , : | The position of the worst frog in the memeplex
: | The adjustment of the th decision variable
: | A random number
: | The maximum adjustment of the th decision variable
: | The renewed position of the th decision variable
: | The upper limits of the position of the th decision variable
: | The lower limits of the position of the th decision variable
: | The total number of global iterations
: | The coding of the decision variable ; it represents the dispatching strategy of , at decision-making instant
: | The probability distribution matrix of the frog population
: | The element of the matrix ; it represents the probability that the th decision variable is valued as
: | The forgetting factor.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work is supported by Funding for Scientific Research Startup of North China University of Technology.

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