Optimization Algorithms Combining (Meta)heuristics and Mathematical Programming and Its Application in Engineering
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An Improved Shuffled Frog Leaping Algorithm and Its Application in Dynamic Emergency Vehicle Dispatching
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 realtime and timedependent link travel speeds are fused, and a timedependent polygonalshaped 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 realtime 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 8^{15}, 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 NPhard 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 realtime traffic data at each decisionmaking instant. They suppose traffic data remain unchanged in the decisionmaking 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 timedependence function, this network is a kind of timedependence network. Research on the timedependence 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 timedependence network. Iteration method was used to solve the shortest path problem. Kaufman and Smith [16] proved that the timedependence shortest path problem can be solved by polynomials only when the network satisfies the firstinfirstout (FIFO) property. Most of the research on the shortest path of timedependence 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 nonFIFO and FIFO network. Ichoua et al. [19] replaced link travel time with link travel speed to build timedependent 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 decisionmaking 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, realtime data (reflecting the realtime road condition at decisionmaking 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 realtime travel speed. Multipleincident and multipleresponse (MIMR) emergency vehicle dispatching discussed in this paper is NPhard problem with large scale variables. The metaheuristic algorithm has advantages in solving the NPhard 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 realword problems such as job shop scheduling and cloud computing resource allocation [27–29].
According to the above, in this paper, the polygonal timedependent function based on realtime 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 timedependent 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 timedependent 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.
(1) Link Travel Time Function. Taking as the minimum time interval, is divided into discrete time intervals, that are , , . is the initial dispatching decisionmaking 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 polygonalshaped travel speed function is shown in Figure 2. At the decisionmaking instant , the realtime link travel speed is known. However, the realtime link travel speed , , , , cannot be obtained. Therefore, they are approximated by the prediction travel speeds .
The polygonalshaped 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 isin 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 realtime 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 polygonalshaped 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 realtime link travel speed.
3. Dynamic Emergency Vehicle Dispatching Modeling
At the decisionmaking 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 decisionmaking instant , every accident needs to be rescued. At other decisionmaking 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 decisionmaking instant , if the emergency vehicle , is dispatched to the accident , then ; otherwise, .
Formulas (12) are constraints for the state of variables . At the decisionmaking 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 decisionmaking 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)
4.1.1. Encoding and Decoding
Decision variables of the emergency vehicle dispatching are 01 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 decisionmaking 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.

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.
(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 isin 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 polygonalshaped travel speed function. Thirdly, we simulated a scene of a new accident to demonstrate the efficacy of adjustment mechanism of dispatching strategy.
