Accelerated Particle Swarm Optimization to Solve Large-Scale Network Plan Optimization of Resource-Leveling with a Fixed Duration

Large-scale network plan optimization of resource-leveling with a fixed duration is challenging in project management. Particle swarm optimization (PSO) has provided an effective way to solve this problem in recent years. Although the previous algorithms have provided a way to accelerate the optimization of large-scale network plan by optimizing the initial particle swarm, how to more effectively accelerate the optimization of large-scale network plan with PSO is still an issue worth exploring. The main aim of this study was to develop an accelerated particle swarm optimization (APSO) for the large-scale network plan optimization of resource-leveling with a fixed duration. By adjusting the acceleration factor, the large-scale network plan optimization of resourceleveling with a fixed duration yielded a better result in this study than previously reported. Computational results demonstrated that, for the same large-scale network plan, the proposed algorithm improved the leveling criterion by 24% compared with previous solutions. APSO proposed in this study was similar in form to, but different from, particle swarm optimization with contraction factor (PSOCF). PSOCFdid not have as good adaptability as APSO for network plan optimization. Accelerated convergence particle swarm optimization (ACPSO) is similar in form to the APSO proposed in this study, but its irrationality was pointed out in this study by analyzing the iterative matrix convergence.


Introduction
The network plan is considered by the engineering community as a promising management method.A large-scale network plan with many works (such as more than 50) is an effective tool for solving large project management problems [1,2].However, the number of possible solutions in largescale network plan optimization sharply increases with the number of works, and the time of calculation is exponential, far beyond the processing capacity of computing resources, so mathematics and computer science cannot solve the problem known as NP problem [2,3].In recent years, genetic algorithm [4,5], Monte Carlo partition optimization [6], and particle swarm optimization (PSO) [7,8] have provided an effective means to solve this problem.
Accelerated optimization can be marked by betteroptimized solutions with the same number of iterations for iterative optimization.Yang et al. introduced some virtual particles in random directions with random amplitude to enhance the explorative capability of particles in PSO [23]; Qi et al. hybridized an improved estimation of distribution algorithm (EDA) using historic best positions to construct a sample space with PSO both in sequential and in parallel to improve population diversity control and avoid premature convergence for optimization of a water distribution network [24]; Zhang et al. added the random velocity operator from local optima to global optima into the velocity update formula of constriction particle swarm optimization (CPSO) to accelerate the convergence speed of the particles to the global optima and reduce the likelihood of being trapped into local optima [25]; Zhou et al. adjusted random functions with the density of the population so as to manipulate the weight of cognition part and social part and executed mutation on both personal best particle and group best particle to explore new areas [26].Zhang and Yang accelerated the optimization of large-scale network plan resources and analyzed the acceleration optimization mechanism via stochastic process by optimizing the initial particle swarm using the Monte Carlo method under limiting conditions [7,8,27]; Ren and Wang proposed a PSO algorithm with accelerated convergence, theoretically proved the fast convergence of the algorithm, and optimized the parameters in the algorithm [28].
Inspired by previous efforts [28] to accelerate the convergence of PSO algorithm, this study proposed the method for the large-scale network plan optimization of resourceleveling with a fixed duration through debugging acceleration coefficient (it might also be described as accelerated PSO, or APSO for short) and yielded a better solution than reported in the previous literature.
This paper is organized as follows.Section 2 describes the experimental research of the large-scale network plan optimization of resource-leveling with a fixed duration using APSO.Section 3 analyzes the difference between APSO and PSO with a contraction factor (PSOCF) [29].Section 4 analyzes the irrationality of accelerated convergence PSO (ACPSO) reported in [28].

APSO to Solve the Large-Scale Network Plan Optimization of Resource-Leveling with a Fixed Duration
Large-scale network plan optimization of resource-leveling with a fixed duration achieved the balance of resource demand in each period during the entire period of the project.Equilibrium could be marked by the variance of resources.
The formula used to calculate the variance was as follows: where the total number of samples   is , and the arithmetic mean of   is .The smaller the variance, the more balanced the resource.
The evolutionary equation of basic PSO was as follows: where  is the number of iterations;   ( + 1) is -dimension space coordinates of particle  in  + 1 iteration;   () is dimension space coordinates of particle  in  iteration;  is inertial factor, usually taking the value of 1 according to the experience; V  () is the -dimension flight speed of particle ;  1 and  2 are accelerators evaluated usually between 0 and 2 by experience; rand 1 and rand 2 are random functions of value in the range of [0, 1];   () is the best place to be experienced by particle ; and   () is the best place for all particles to experience.The convergence condition was adopted by setting maximum iteration times .
The evolutionary equation of accelerated PSO (APSO) was as follows: where  is the acceleration coefficient, and the other signs are the same as earlier.The evolution equation of accelerated particle swarm algorithm has one more  than with that of the basic PSO algorithm and one more  than that of the particle swarm algorithm with contraction factor.However, it has produced significant results for solving large-scale network plan optimization of resource-leveling with a fixed duration as follows.
For example, a large network plan with 223 works is the same as Figure 1 in [27].The debugging results of change a are shown in Table 1, where the variance of the corresponding optimization results is 17.58 (better than the variance 22.99 quoted in [27]).The start time of each work is shown in Table 2, and the resource requirements of each unit time are shown in Table 3.
As shown in Table 1, for  = 1,  1 =  2 = 2.05, the number of particles 50, and  = 100, the minimum variance 17.58 could be obtained by adjusting the acceleration coefficient , which was significantly optimized compared with the variance quoted in [27] without the acceleration coefficient (that is 22.99).For  = 1,  1 = 3.5,  2 = 0.4, the number of particles 50, and  = 100, the minimum variance 18.4 could be obtained by adjusting the acceleration coefficient , which was significantly optimized compared with the variance quoted in the literature [27].For  = 0.8,  1 =  2 = 2.05, the number of particles 50, and  = 100, the minimum variance 18.93 could be obtained by adjusting the acceleration coefficient , which was significantly optimized compared with the variance quoted in [27].For  = 0.729,  1 =  2 = 1.454, the number of particles 50, and  = 100, variance smaller than 17.83 (acceleration coefficient 1) could not be obtained by adjusting the acceleration coefficient .

Difference between APSO and PSOCF [29]
APSO proposed in this study was similar in form to PSOCF.The evolution equation of PSOCF was as follows [29]: where contraction factor The other signs are the same as earlier.
For  1 = 3.5,  2 = 0.4,  = 2/|2−−√( − 4)| = 2/|2− 3.9 − √3.9(3.9 − 4)| does not exist.The PSOCF could not be used, but APSO in this study was used for optimization of network plan and the results were good, as shown in Table 1. For The acceleration factor  is outside the scope of the contraction factor , and the optimization of APSO in this study was performed as usual, as shown in Table 1.
Thus, although, in this study, APSO was similar in form to PSOCF, essentially, for network plan optimization, PSOCF did not have as good adaptability as APSO.

Irrationality of ACPSO Reported in [28]
APSO proposed in this study was inspired by the ACPSO algorithm quoted in [28].APSO was similar in form to ACPSO.The evolution equation of ACPSO algorithm proposed in [28] was as follows: where  is angle value with a distinct optimization effect when its value is within [0, /8];  is a constant greater than zero, and the optimization effect is good when the value is 3.The other signs are the same as earlier.
The ACPSO algorithm quoted in [28] was based on one inference: PSO is iterative.The iterative converges when the spectral radius of iterative matrix  (that is the maximum absolute value of the matrix eigenvalue) is less than 1.The smaller the spectral radius of , the faster the iteration  The evolution equation of PSO can be written in the matrix form: where , is the best place ever found, and  , is the best location for the whole particle swarm to date.The other signs are the same as earlier. Set, Then, ( 5) can be changed to is the mathematical expectation.
The following two equations are equivalent to the inference in [30]: where  is the coefficient matrix,  is the unknown column vector,  is a constant number column vector, and  is a constant matrix determined by  and .
The following iterative matrix  could be obtained by the aforementioned system: Set  * as the solution of the system.Then, The aforementioned two formulas ( 13) and ( 14) on subtraction yield Because  0 −  * has nothing to do with , lim →∞ ( +1 −  * ) = 0 is equivalent to lim →∞  +1 = 0.The theorem quoted in [30] shows lim →∞  +1 = 0 equivalent to () < 1, where () is the spectral radius of matrix .
Thus, the iterative matrix did not necessarily converge.Because the particle swarm algorithm did not have a set of equations to solve  +1 , the aforementioned reasoning could not be executed using the iterative matrix  convergence.

Conclusions
This study proposed the method for the large-scale network plan optimization of resource-leveling with a fixed duration through adjusting the coefficient of APSO based on the algorithm quoted in [27] to obtain a better solution than previously reported.In other words, for the same large-scale network plan, the proposed algorithm improved the leveling criterion by 24% compared with previous solutions.Thus, the resource variances of 17.58 and 223 of a large-scale network plan are the best results for the large-scale network plan optimization of resource-leveling with a fixed duration to date in the literature.
Section 3 discusses the difference between APSO proposed in this study and PSOCF quoted in [29].The proposed APSO was similar in form to PSOCF, but, essentially, PSOCF did not have as good adaptability as APSO for the network plan optimization.
Section 4 describes the difference between APSO proposed in this study and ACPSO quoted in [28].Through Mathematical Problems in Engineering analyzing the iterative matrix convergence of equations, it was pointed out that the derivation of iterative matrix convergence of ACPSO algorithm proposed in [28] was problematic, although it experimentally proved APSO was similar in form to ACPSO.
The effect of the APSO proposed in this study was verified to be obvious experimentally.However, the internal working mechanism of APSO is still a core issue worth investigation.

Table 1 :
Optimization parameter's debugging results of the large-scale network plan optimization of resource-leveling with a fixed duration using the accelerated particle swarm algorithm (particle number is 50;  = 100).
Note.The value in italics is the optimal value under certain parameter conditions.

Table 2 :
The parameters and their optimization solution for the optimization example of the resource-leveling with a fixed duration.
Note. "ES" is the early start time of each work."Optimized ES" is the start time optimized.