Improved Particle Swarm Optimization for Selection of Shield Tunneling Parameter Values

: This article proposes an exponential adjustment inertia weight immune particle swarm optimization (EAIW-IPSO) to enhance the accuracy and reliability regarding the selection of shield tunneling parameter values. According to the iteration changes and the range of inertia weight in particle swarm optimization algorithm (PSO), the inertia weight is adjusted by the form of exponential function. Meanwhile, the self-regulation mechanism of the immune system is combined with the PSO. 12 benchmark functions and the realistic cases of shield tunneling parameter value selection are utilized to demonstrate the feasibility and accuracy of the proposed EAIW-IPSO algorithm. Comparison with other improved PSO indicates that EAIW-IPSO has better performance to solve unimodal and multimodal optimization problems. When solving the selection of shield tunneling parameter values, EAIW-IPSO can provide more accurate and reliable references for the realistic engineering.


Introduction
Shield tunneling method is widely used in the construction of urban subway tunnels due to its small disturbance to the surrounding environment and low construction cost. Shield tunnel construction is located in the underground space, and the main engineering part relies on the shield machine for construction. Therefore, the control of the ground settlement has a great influence on the shield tunneling. Effective control of ground settlement can ensure smooth construction and requirements of the period. Ground settlement is affected by many factors, including three major categories: factors that cannot be changed (hydrology, geology), factors that are less controllable after the scheme is determined (tunnel radius, shape), and controllable factors [Preisig, Dematteis, Torri et al. (2014)]. Among the controllable factors, the selection of shield tunneling parameters is the key to control ground settlement [Bouayad, Emeriault and Maza (2015)]. Selecting better shield tunneling parameter values can reduce the interference of construction on the land, control ground settlement value effectively and improve construction efficiency. Many researchers [Moeinossadat, Ahangari and Shahriar (2017); Zhou, Ding and He strategy. However, the big inertia weight makes the algorithm have a strong global search capability, which may destroy the search of the global optimal value for multimodal functions and obtain the local optimum in the end. Han et al. [Han, Li, and Wei (2006)] adjusted the inertia weight according to the fitness value of each particle and the premature condition of the particle group. However, the local optima will have a greater impact on the adjustment of other particles under this adjustment strategy. It will be easy to fall into local optima for multimodal function. Feng et al. [Feng and Liu (2016)] adopted PSO with exponentially decreasing inertia weight to solve non-differentiable NPhard problem of absolute value equations. Adjusting the inertia weight in exponentially decreasing form improves the convergence speed to some extent, but the particle diversity in the later iterations cannot be guaranteed and algorithm still cannot obtain a better global optimal value. Studies find that the improved PSO algorithms based on dynamic adjustment inertia weight from former researchers are also easy to fall into the local optima. The global search ability is still insufficient. In addition, particles adjust themselves according to the surrounding particle positions. The dependence between the particles is large and the particle group lacks the adjustment mechanism, which is also one reason why PSO is easy to fall into the local optima. In order to enhance the accuracy and reliability regarding the selection of shield tunneling parameter values based on PSO, the new adjustment equation of the inertia weight is given first. Meanwhile, the self-regulation mechanism of the immune system is combined with PSO. 12 benchmark functions are applied to test the performance of EAIW-IPSO. Then, EAIW-IPSO is applied to solve the selection of shield tunneling parameter values. The realistic cases of shield tunneling parameter optimization are studied at the end. The rest of the paper is organized as follows: The second section gives basic PSO theory and improvement analysis. The third section specifically improves PSO and proposes EAIW−IPSO. The fourth section proves the performance of EAIW-IPSO based on 12 benchmark functions. The fifth section constructs the optimization method of the tunneling parameters based on EAIW-IPSO. Realistic cases for shield tunneling parameter optimization are studied in the sixth section. Conclusions are presented in seventh section.

The basic PSO theory and improvement analysis
PSO is one of the computational intelligence methods, which is a process of simulating bird foraging [Sedghizadeh and Beheshti (2018)]. Each bird is assumed as a particle in PSO. Particle velocity and position are updated by two optimal positions [Chen, Li, Xiao et al. (2018)]. One of the optimal positions is the personal best position and the other optimal position is the global best position. Particles are extended to N-dimensional space. The size of particle group is M. The position of particle i is expressed as vector Xi=(xi1, xi2, …, xiN) and velocity is expressed as vector Vi= (vi1, vi2, …, viN). The velocity and position update according to the Eqs. (1) and (2) [Nie, Wang, Xiao et al. (2017)].
where ω is inertia weight, c1 and c2 are acceleration constants, r1 and r2 are random values within the range of (0,1), pis is personal best position of the ith particle in sth direction (i＝1, 2, …, M) (s＝1, 2, …, N), pgs is global best position in sth direction. Many researchers have studied the inertia weight in Eq. (1) [Chatterjee and Siarry (2006); Miao, Shi, Zhang et al. (2009);Pluhacek, Senkerik, Davendra et al. (2013); Taherkhani and Safabakhsh (2016); Uma, Gandhi and Kirubakaran (2012)]. Inertia weight is initially seen as a constant value, but subsequent studies have found that changing the inertia weight value has an impact on algorithm performance. The adjustment equation for inertia weight that was first adopted is as follows: where ωmin is minimum value of inertia weight, ωmax is maximum value of inertia weight, t is current number of iterations, T is maximum number of iterations. In Eq. (3), the inertia weight is linearly decremented during the iterative process. With the deepening of the research, however, researchers have analyzed and concluded that the change of inertia weight should be a nonlinear adjustment in the iterative process. Eq. (4) applies random function to adjust the inertia weight nonlinearly.
Eq. (5) and Eq. (6) use the exponential form to dynamically adjust the inertia weight, which is also a typical nonlinear adjustment strategy.
where d1, d2 are control constants. According to the curve of exponential decreasing function, it can be found that the curve of exponential decreasing function is consistent with the iterative process of PSO. Therefore, the form of exponential decreasing is selected to adjust the inertia weight first. Meanwhile, this article considers the range of inertia weight and the change of inertia weight in the iterative process. In addition, particles adjust themselves according to the surrounding particle positions and lack the mechanism of variation, which may lead to poor particle diversity [Chen, Cao, Ye et al. (2013)]. In order to solve such problems, the self-regulation mechanism of the immune system will be used to select the next generation of group. max min min In Eq. (7), when random number is 0, ω is ωmin. ω will be ωmax when random number is 1. In addition, the study results show that the global search ability is strengthened when the inertia weight is big and the local search ability is strong when the inertia weight is small. The inertia weight should be dynamically adjusted during the iterative process. In the early stage to search for the global optimal value in a large range, the global search ability should be strengthened. At this time, the inertia weight should be big. The inertia weight should be reduced in the late iterations, and the optimal value search should be performed in the local range. Therefore, the inertia weight should be reduced as the number of iterations increases. In order to improve the speed of convergence to the vicinity of the global optimal value in the early iterations, and then perform local search near the optimal value, Eq. (7) is modified as follows according to the above analysis: In Eq. (8), the value of t is small in early stage, the value of (1-(t/T) 2 ) will be big. Due to the value of (ωmax/ωmin) is over 1, ω will be big according to the monotonicity of exponential function. In the late iterations, the value of (1-(t/T) 2 ) will decrease with the increase of t, so that ω will decrease. Moreover, to further reduce the possibility of falling into local optima in the search process, the article selects the next generation group based on the self-regulation mechanism of the immune system. The specific selection strategy is: before the end of the each iteration, another particle group with size of M is initialized and the fitness value of the corresponding particle is calculated. Updated and initialized particles form a group with size of 2M. Then, M particles with relatively large fitness value in particle group with size of 2M are selected as the next generation particle group. The steps and procedures of the EAIW-IPSO are realized as follows: Step1: Set the size of particle group M, maximum number of iterations T, acceleration constants c1 and c2, minimum and maximum values of inertia weight ωmin and ωmax.
Step3: Update the position and velocity values based on Eqs.
(1), (2) and (8), calculate the fitness value of each updated particle, update the global best position Pg and the personal best position Pi (i＝1, 2, …, M). Step4: Initialize another particle group with size of M, calculate fitness value and set the personal best position Pi (i＝1, 2, …, M) for initialized particles.
Step5: Sort the fitness values about 2M particles in updated and initialized group, select the M particles with relatively large fitness value as next iteration group.
Step6: If the termination condition is satisfied, the iterations stop. Otherwise, the next iteration is entered.
4 Experimental study 12 benchmark functions in Tab. 1 are applied to test the performance of the proposed EAIW-IPSO algorithm. f1, f2, f3, f7, f8, f9 are unimodal with only one optimum and the others are multimodal with some local optima. All functions obtain the standard optimal value of 0. Meanwhile, EAIW-IPSO has also been tested against with PSO, linear decreasing inertia weight particle swarm optimization (LDIW-PSO), random inertia weight particle swarm optimization (RIW-PSO), and exponentially decreasing inertia weight particle swarm optimization (EDIW-PSO). Experiment parameters are set as follows: the minimum and maximum values of inertia weight are 0.4 and 0.9, the maximum number of iterations is 100, the acceleration constants c1 and c2 are 2, and the size of particle group is 50.  The five algorithms run 20 times independently in each function. The indicators to evaluate the performance of the algorithm are: the best value of 20 run, the mean value, and the root mean square error (RMSE). Among these indicators, the best and mean values are used to evaluate the optimization accuracy of the algorithm, and RMSE is to evaluate the optimization stability. The simulation results are shown in Tab. 2. Meanwhile, all best and mean values have been drawn as line charts.

Unimodal function
The optimization performance of the five algorithms in unimodal function can be analyzed according to the simulation results of f1, f2, f3, f7, f8, f9. As shown in Tab. 2, the best and mean values of PSO are all the biggest in five algorithms. The best and mean values of LDIW-PSO, EDIW-PSO, and RIW-PSO are smaller than PSO respectively, but these algorithms cannot solve these unimodal functions all with good accuracy. EAIW-IPSO obtains the best solutions among these algorithms. In addition, it also can be seen from Fig. 1  In Fig. 2, the graphical results show the changing curve of the fitness value with the iterations of different algorithms for f1, f2, f3, f7, f8, f9. As shown in Fig. 2, LDIW-PSO, EDIW-PSO, and RIW-PSO have a faster convergence rate, but their global search values at the early stage are worse than the value of EAIW-IPSO. Compared with other algorithms, EAIW-IPSO can quickly converge to a more accurate value, and then converge to a better solution with the local search in the late iterations. Therefore, the convergence characteristics show that EAIW-IPSO has a stronger search capability and can obtain more accurate global optimal value for unimodal functions.

Multimodal function
The functions of f4, f5, f6, f10, f11, f12 are multimodal with some local optima. It can be seen from Tab. 2 that the optimization values obtained from LDIW-PSO, EDIW-PSO, and RIW-PSO are close, which shows that these three algorithms have the relatively close level in accuracy for multimodal functions. According to the positions corresponding to multimodal functions in lines of best and mean in Fig. 1, EAIW-IPSO does not show particularly good accuracy in f11. But the optimization values of EAIW-IPSO are much better in other multimodal functions. The stability of EAIW-IPSO is also better than that of other algorithms. Therefore, EAIW-IPSO still has high accuracy in multimodal functions. The iteration processes of f4, f5, f6, f10, f11, f12 are shown in Fig. 3. The graphical results show the changing curve of the fitness value with the iterations of different algorithms for f4, f5, f6, f10, f11, f12. As can be seen, EAIW-IPSO can converge to a more accurate value quickly at the early stage too. The search ability is still strong to obtain a better global optimal value in the late iterations, which shows that EAIW-IPSO has better performance to overcome the problem of falling into the local optima. According to the performance analysis of five algorithms in unimodal and multimodal functions, EAIW-IPSO has better accuracy and convergence characteristics. EAIW-IPSO integrates the advantage of maintaining the particle diversity through self-regulation mechanisms in immune algorithm. Therefore, the iteration process allows for a wider range of search, reduces the influence of local optimal particle position on other particles and has better ability to gain global optimal value. Moreover, EAIW-IPSO has better convergence characteristic of other improved PSO by adjusting inertia weight in exponential form. The PSO that uses the exponential form to adjust the inertia weight can quickly converge to the position of the global optimal value in the early iterations, which can improve the iterative efficiency of the algorithm. In summary, the optimization values obtained from EAIW-IPSO are more accurate and stable compared with other four algorithms, regardless of the unimodal or multimodal optimization problems. EAIW-IPSO has better overall performance to search the global optimal value.

Shield tunneling parameter optimization based on EAIW-IPSO
Before using EAIW-IPSO to optimize the shield tunneling parameters, it is necessary to construct the relationship model between the tunneling parameters and the ground settlement. To better predict the nonlinear relationship between the tunneling parameters and ground settlement in realistic projects, the article also considers the geometric and formation condition parameters. BP neural network optimized by genetic algorithm (GA-BP) is applied to construct the nonlinear relationship prediction model between selected engineering parameters and ground settlement. Based on realistic data, the neural network prediction model is trained. The final weights and thresholds of the trained neural network could be obtained. The neural network model to predict the relationship between selected engineering parameters and ground settlement can be constructed as follows: where wij is obtained weight between input layer and hidden layer; hj is obtained threshold of hidden layer; wj' is obtained weight between hidden layer and output layer; h is obtained threshold in output layer; L is number of nodes in hidden layer; ki is the ith selected engineering parameter; q is the number of engineering parameters; f(x) is activation function. Then, EAIW-IPSO is adopted to optimize the tunneling parameters under specific geometric and formation conditions based on predictive model.
The following values need to be set: the maximum number of iterations H, acceleration constants c1 and c2, minimum and maximum values of inertia weight ωmin, ωmax, and particle group size M. According to the selected engineering parameters, the particle space dimension is q. The selected engineering parameters need to be initialized as the position of particle Xi＝ (xi1, xi2, …, xiq) ＝(k1i, k2i,…, kqi) (i＝1, 2, …, M). In addition, the late change values of the selected engineering parameters need to be initialized as the particle velocity Vi ＝ (vi1, vi2, …, viq) (i＝1, 2, …, M). The important thing to note is that the geometric and formation condition parameter values should be constant during the optimization process. Each particle fitness value is calculated based on the obtained predictive model. The fitness value of particle I(I＝1, 2, …, M) is: The initialized position for each particle is taken as the corresponding personal best position Pi＝(pi1, pi2, …, piq) ＝(xi1, xi2, …, xiq) (i＝1, 2, …, M). The position of gth particle with maximum fitness value are considered as global best position Pg＝(pg1, pg2, …, pgq) ＝(xg1, xg2, …, xgq). Particle velocity and position are updated by Eqs. (1) and (2). Meanwhile, Eq. (8) is adopted to calculate inertia weight. Then, the updated particle fitness values are calculated again. The personal best position of each particle and global best position are updated.
Other M particles are initialized and corresponding fitness value is calculated too. The personal best position of each initialized particle is set. And then, all particle fitness values in updated and initialized groups are sorted. The M particles with relatively large fitness value are selected as the next iteration group. Meanwhile, the personal best position of particle and global best position are updated. Cycling iteration process until the number of iterations is over.
6 Case study for shield tunneling parameter optimization 6.1 Case 1 Shield tunneling parameter optimization of Changsha metro line 1 is selected as the study case. The tunneling parameters that affect the ground settlement considered in this case are: synchronous grouting amount, shield thrust, cutter head torque, the ratio of tunneling speed and cutter speed R (the cutter speed is usually 1.5 rad/min), and earth pressure. As the same time, geometric condition parameter is: the ratio of buried depth H and diameter excavation D. Formation condition parameters are: groundwater level, cohesion, internal friction angle, earthwork heavy, and side-pressure coefficient. Since the main objective of the study is to find the best tunneling parameter values to minimize the ground settlement, the 34 groups of samples [Mou (2013) (2016)]. The training performance of predictive model is shown in Fig. 4. Tab. 4 shows the test results.

Case 2
Changsha-Zhuzhou-Xiangtan Intercity Railway is selected as the second case study. The tunneling parameters considered in this case are: synchronous grouting amount, shield thrust, cutter head torque, the ratio of tunneling speed and cutter speed R, earth pressure, and slag amount. Meanwhile, geometric condition parameter is: the ratio of buried depth H and diameter excavation D. Formation condition parameters are: groundwater level and earthwork heavy. The sample data are shown in Tab. 8.    The obtained ground settlement values are shown in Tab. 11. The results also prove the better performance of EAIW-IPSO. As can be seen in Tab. 11, the ground settlement value after optimized is close to 0, which is an ideal value. In realistic engineering, the existence of many influencing factors will lead to the ground settlement. Hence, the optimized tunneling parameter values should be used to test its realistic ground settlement value first. And then, the final adopted tunneling parameter values in realistic engineering are further adjusted according to the optimized values. Tab. 12 shows the optimized tunneling parameter values based on EAIW-IPSO and adjusted tunneling parameter values in this case. Tab. 12 also illustrates that the obtained parameter values based on EAIW-IPSO can provide better references for selection of realistic tunneling parameter values. Therefore, tunneling parameters are optimized based on EAIW-IPSO first in realistic projects, and then further adjusted according to the optimized values, which will improve the efficiency of the tunneling parameter value selection.

Conclusions
The selection of shield tunneling parameter values has a very important influence on tunnel construction. To provide more accurate and reliable references for selection of shield tunneling parameter values based on PSO, the article proposes EAIW-IPSO. In proposed algorithm, the new adjustment equation for inertia weight is given, and selfregulation mechanism of the immune system is combined with the PSO. EAIW-IPSO has been tested against with PSO, LDIW-PSO, RIW-PSO, EDIW-PSO based on 6 unimodal functions and 6 multimodal functions. Simulation results indicate that EAIW-IPSO can converge to a more accurate value at the early stage quickly, and local search ability is still strong in the late iterations. In addition, the proposed algorithm has better performance to overcome the problem of falling into the local optima. The optimization method of shield tunneling parameters is constructed based on the proposed EAIW-IPSO. GA-BP neural network is applied to construct the predictive model between the selected engineering parameters and ground settlement first. EAIW-IPSO is used to optimize tunneling parameters under specific geometric and formation condition based on predictive model. To evaluate the performance of the proposed algorithm in selecting the shield tunneling parameter values, two realistic cases are studied. The case results verify the feasibility and accuracy of EAIW-IPSO. Therefore, EAIW-IPSO can improve the efficiency for the selection of tunneling parameter values in realistic engineering.