Crowd dynamics is a discipline dealing with the management and flow of crowds in congested places and circumstances. Pedestrian congestion is a pressing issue where crowd dynamics models can be applied. The reproduction of experimental data (velocitydensity relation and specific flow rate) is a major component for the validation and calibration of such models. In the social force model, researchers have proposed various techniques to adjust essential parameters governing the repulsive social force, which is an effort at reproducing such experimental data. Despite that and various other efforts, the optimal reproduction of the real life data is unachievable. In this paper, a harmony searchbased technique called HSSFM is proposed to overcome the difficulties of the calibration process for SFM, where the fundamental diagram of velocitydensity relation and the specific flow rate are reproduced in conformance with the related empirical data. The improvisation process of HS is modified by incorporating the global best particle concept from particle swarm optimization (PSO) to increase the convergence rate and overcome the high computational demands of HSSFM. Simulation results have shown HSFSM’s ability to produce near optimal SFM parameter values, which makes it possible for SFM to almost reproduce the related empirical data.
Massive congestion in pedestrian environments has been a pressing concern lately. This issue stems from an increase in population growth rate, where among the serious problems linked to it are crowd stampedes, accidents, and other disasters. Congestion has therefore motivated researchers to find solutions, where studies have focused on offering good pedestrian facilities or understanding incorrect pedestrian behavior and correcting them [
May [
For the purpose of validation of such microscopic crowd dynamics models, a variety of methods was adopted. These methods include the introduction of the selforganization phenomena and the reproduction of experimental data obtained by experimental and empirical studies. The experimental data, for instance, are the specific flow rate (number of persons crossing an opening per unit of time and width) and the fundamental diagrams (a graphical representation as shown in Figure
The diagram for the estimated speeddensity relations for unidirectional pedestrian traffic flow in empirical studies. Fruin [
The main concern is the calibration of these models. A common method of the calibration process involved with the models’ assessment is to optimize some model parameters and model components long enough; until, the simulations fit the fundamental diagram, such as that of Weidmann [
In this paper, we focus our attention on proposing new technique for calibrating essential parameters in the social force model (SFM). The model has been rendered as one of the most vital models in microscopic studies because of its successful introduction of selforganization phenomena of pedestrian dynamics in normal and panic situations [
In [
In this paper, a harmony searchbased optimization technique named HSSFM is proposed to overcome the difficulties of the calibration process for the SFM, wherein the fundamental diagram of velocitydensity relation and the specific flow rate are reproduced in conformance with the related empirical data. Harmony search (HS) is a relatively new population based on metaheuristic algorithm, which has obtained excellent results in the field of combinatorial optimization [
The organization of this paper is described as follows. Section
The representation of the pedestrians’ motivations as social forces implemented in a Newtonian equation for introducing the pedestrians’ motion is an essential feature of the SFM. An extension by Helbing et al. [
The main equations of the model are
The model of the repulsive social force is
Geem et al. [
In the case of HS, optimization is treated as a minimization (or maximization) problem. Basically, minimize (or maximize)
The harmony memory (HM) is a matrix of solutions with size HMS, as shown in (
In this step, the HS generates (improvises) a new harmony vector,
The variable
The generated harmony vector,
Termination occurs when the maximum number of improvisations NI is reached.
The following section describes the proposed HSSFM algorithm.
This section presents the proposed HSSFM algorithm in detail and shows how HS is implemented to find the optimal values of SFM’s parameters.
Each harmony memory vector represents a candidate of SFM parameter values. To initialize the HM with feasible solutions, each parameter value is randomly generated from its valid range. In this study, the valid ranges are as follows:
strength of the social repulsive force
repulsive distance range of the corresponding force (
angular parameter
For example, the vector
In order to discover the goodness of the generated harmony vector with regard to SFM, the fitness function for each harmony vector is calculated (as will be discussed in Section
The new harmony vector is a new candidate SFM parameter value vector, and the values of this new harmony vector is generated depending on the HS’s improvisation rules. In this study, however, memory consideration, which is the most vital operator in HS, is modified. In the standard HS, most of the variables in the new harmony are randomly selected from the other vectors stored in HM. In this study, we adopt the supporting mechanism concept from PSO [
Based on this modification, the values of the new harmony vector components will be selected from the best harmony memory vector stored in HM, with the probability of HMCR. On the other hand, the value of the components of the new harmony vector is selected from the possible range with a probability of 1HMCR. The following equation summarizes these two steps, that is, memory consideration and random consideration
In order to measure the goodness of each harmony memory vector, which in turn measures the goodness of the SFM parameter values, a new fitness function is proposed that includes the two main factors that affect the goodness of the solution with respect to SFM. These factors are
the reproduction of the fundamental diagram and
the reproduction of the flow rate.
The basic model of our proposed fitness function is
The measurement flow rate is defined as the number of pedestrians passing through a section per unit time. The allowed range for a specific flow rate is [low flow, up Flow], which is determined according to the relevant empirical studies mentioned in [
This section presents the simulation results of the proposed HSSFM algorithm. All simulations were conducted using MATLAB version R2010a. Each simulation comprises two scenarios (or parts), which were conducted simultaneously. The physical environment in the first scenario used for the reproduction of the fundamental diagram is that used in [
HSSFM parameters.
The Pedestrian’s parameters  


The pedestrians’ mass: uniformly distributed within the range 

The pedestrians' radius: uniformly distributed within the range 


The SFM’s parameters  



The strength of the contact (pushing) force 

The strength of the friction 

The preferred velocity 

The reaction time 

The fluctuation source of the pedestrian’s acceleration is randomly assigned to each individual 


The HSSFM parameters  


HMCR = 0.95  Harmony memory consideration rate 
HM = 20  Harmony memory size 
PAR = 0.75  Pitch adjustment rate 
NI = 500  Number of improvisation 
The validation of each simulation being done by HSSFM algorithm is done by a comparison between those results obtained from HSSFM algorithm and those obtained from the fundamental diagram, developed by Weidmann [
The simulations were run 400 times, and a snapshot of the results is shown in Table
HSSFM simulation results.
Simulation _number 


Angular  Density 1  Density 2  Density 3  Density 4  Density 5  Density 6  Flow rate 100  Flow rate 200  Flow rate 300  Flow rate 400 





















13  230.80  0.61  0.74  1.07  0.71  0.36  0.02  0.02  0.02  1.91  2.09  2.22  2.30  0.09  0.61  0.70 

















21  1705.85  0.87  0.97  1.00  0.63  0.24  0.04  0.03  0.09  0.41  1.84  2.95  3.84  0.18  3.58  3.76 
22  234.92  0.90  0.80  1.04  0.68  0.32  0.02  0.02  0.02  1.82  2.64  2.10  2.05  0.13  0.79  0.91 

















36  66.16  0.46  0.02  1.11  0.75  0.36  0.02  0.02  0.02  0.74  1.02  1.28  1.46  0.17  0.64  0.81 
37  751.33  0.65  0.92  1.07  0.72  0.36  0.02  0.02  0.02  1.50  3.08  3.61  4.20  0.09  4.88  4.97 

















51  448.75  0.42  0.81  1.10  0.73  0.30  0.02  0.02  0.02  2.16  2.72  2.38  3.29  0.20  2.56  2.77 
56  216.29  0.97  0.78  1.01  0.65  0.27  0.02  0.02  0.02  1.96  1.75  2.47  2.91  0.15  1.39  1.54 
57  1025.38  0.81  0.94  1.03  0.67  0.29  0.02  0.02  0.02  0.95  2.39  3.18  3.83  0.16  3.65  3.81 
62  1597.78  0.80  0.97  1.05  0.71  0.36  0.04  0.03  0.03  0.62  2.11  3.10  3.72  0.10  3.51  3.61 
91  470.11  0.46  0.84  1.10  0.75  0.37  0.02  0.02  0.02  1.95  2.87  3.25  3.38  0.15  3.50  3.65 

















113  56.93  0.89  0.26  1.06  0.70  0.38  0.02  0.02  0.02  0.74  0.86  1.17  1.33  0.08  0.84  0.91 
120  1687.54  0.66  0.96  1.06  0.72  0.36  0.03  0.02  0.02  0.72  2.23  3.07  3.99  0.10  3.77  3.87 
131  361.66  0.93  0.87  1.05  0.71  0.35  0.02  0.02  0.02  1.55  2.57  3.37  3.87  0.10  3.81  3.91 
145  326.49  0.81  0.85  1.06  0.71  0.36  0.02  0.02  0.02  2.03  2.70  3.11  3.07  0.10  2.91  3.01 
160  293.08  0.66  0.80  1.07  0.70  0.36  0.02  0.02  0.02  1.95  2.39  2.40  2.54  0.09  1.33  1.41 
161  1640.52  0.58  0.96  1.06  0.71  0.34  0.02  0.02  0.02  0.88  2.24  3.18  4.00  0.12  3.73  3.85 
189  907.16  0.59  0.93  1.07  0.73  0.35  0.02  0.02  0.02  1.70  2.90  3.85  4.20  0.10  4.94  5.04 

















208  198.47  0.67  0.71  1.06  0.68  0.34  0.02  0.02  0.02  1.46  1.50  1.86  2.06  0.10  0.06  0.16 

















362  402.80  0.72  0.86  1.06  0.70  0.34  0.02  0.02  0.02  2.02  3.14  3.20  2.98  0.10  3.34  3.44 


































394  1625.67  0.79  0.97  1.04  0.69  0.33  0.04  0.03  0.02  0.65  2.11  3.07  3.91  0.12  3.63  3.75 
For further clarification of the results obtained from the HSSFM algorithm and the results shown in Table
Optimal SFM's parameter ranges.
SFM’s parameter  Optimal range  Means value 



173 


0.69 


0.66 
The fundamental diagrams of our HSSFM simulations compared with Weidmann [
It is important to note that firstly the resulting fundamental diagram shown in Figure
Thirdly, the range of the
Fourth, the repulsive distance parameter in [
The SFM is considered one of the most distinguished representative microscopic dynamics models that present solutions to pedestrians’ congestion problems. In order to improve its performance, we proposed the HSbased algorithm HSSFM that offers near optimal parameter values for the SFM. Simulations conducted showed the ability of HSSFM to find near optimal parameter values that can help in the reproduction of both the fundamental diagram as in [
Our future work will focus on improving the ability of HSSFM to reduce the difference between the mean velocity
As the microscopic dynamics models have been exposed to many amendments, such as the incorporation of new submodels or refining the models itself, the need for a technique that can offer near optimal parameter values is necessary. We foresee that more effort for the development of these techniques is necessary by studying and potentially improving their behaviors in exploring the search space of the dynamics models parameter values. We also recommend the hybridization of techniques such as harmony search with other metaheuristic optimization algorithms in order to improve search capabilities to find the optimal parameter values of the microscopic dynamics models. This in turn can help improve the representation capabilities of the microscopic dynamics models to represent pedestrians’ congestion problems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the Deanship of Scientific Research, University of Tabuk for their financial support (Grant no. S1831434).