Large-Scale Network Plan Optimization Using Improved Particle Swarm Optimization Algorithm

No relevant reports have been reported on the optimization of a large-scale network plan with more than 200 works due to the complexity of the problem and the huge amount of computation. In this paper, an improved particle swarm optimization algorithm via optimization of initial particle swarm (OIPSO) is first explained by the stochastic processes theory. Then two optimization examples are solved using this method which are the optimization of resource-leveling with fixed duration and the optimization of resources constraints with shortest project duration in a large network plan with 223 works.Through these two examples, under the same number of iterations, it is proven that the improved algorithm (OIPSO) can accelerate the optimization speed and improve the optimization effect of particle swarm optimization (PSO).


Introduction
A large-scale network plan that is composed of more than 50 works has become an essential tool for managing large-scale engineering project [1,2].However, due to the rapidity of solution increase (called the combustion explosion) and the exponential growth of computing time with the complexity of the problem, which far exceeds the processing capacity of computing resources, the optimization of large-scale network plan becomes an unsolvable problem in the mathematics and computer science fields, also called the NP problem [3][4][5][6][7].Among the existing optimization methods for network plan, accurate algorithm such as the dynamic planning [8], 0-1 planning [9,10], and branch and bound method [11,12] can solve the small network plan optimization; various heuristic algorithms [13][14][15][16][17][18] cannot solve large-scale network plan optimization.An effective way to solve the complex network plan is by using genetic algorithm (GA), but the works numbers of presented examples (86 and 122) are not large enough [19][20][21][22][23].
The Monte Carlo method can be applied to solve equations, integral equations, difference equations, integral, shielding radioactive particles, neutron fission security problems, the random service (queuing theory) of economic service problems, signal detection and system simulation, flow field simulation, life test, and more [44].As it optimizes initial particle swarm to solve the optimization problem of large-scale network plans, it is PSO's foundation.Zhang and Shi [45] adopted the Monte Carlo method to solve the optimization of the resource-leveling with fixed duration and the resources constraints with shortest project duration of a network plan.But the works number of the presented examples (9) is not large enough.To solve the optimization problem of the resource-leveling with fixed duration in a large network plan, Du et al. [46] proposed partition optimization based on the Monte Carlo method.However, there is not a large enough works number (61), and partition optimization combination may lose the global optimal solution in theory.
Without requiring any advanced knowledge of the reliability function, PSO combining with the Monte Carlo method was used to solve complex network reliability problems, while Monte Carlo method was used to evaluate system reliability, but its motivation is different from this paper [47].In comparison to the random method, the Monte Carlo method in the Monte Carlo Enhanced PSO can calculate the probability of initial particles' elements and form better initial particles, but there was no analyzation of the improvement mechanism to optimize the initial particle swarm [48].To solve resource optimization and cost optimization of a largescale network plan by using PSO, Zhang and Yang used the Monte Carlo method under limited conditions to optimize the initial particle swarm [49,50].However, there was no analyzation of the mechanism improvement to optimize the initial particle swarm, and the works number of presented examples (61) is not large enough.
In this paper, an improved particle swarm optimization algorithm via optimization of initial particle swarm (OIPSO) is first explained by the stochastic processes theory.Then two optimization examples of a large-scale network plan are solved using this method, which are the optimization of resource-leveling with fixed duration and the optimization of resources constraints with shortest project duration in a large network plan with 223 works.The optimization effect of the improved algorithm (OIPSO) is proven through these two examples.
This paper is organized as follows: Section 2 analyzes the improvement mechanism of OIPSO, Section 3 solves large-scale (223 works) network plans by OIPSO, Section 4 introduces the superiority of OIPSO compared with the original and existing PSOs, and Section 5 makes conclusions.

Methodology
2.1.OIPSO.The process of the original PSO is as follows [30]: Step 1. Determining the initial particle swarm Step 2. Evolving the particle location Step 3. Determining each particle's best experiencing position and all particles doing Step 4. Outputting the optimization results when the maximal number of iterations is reached; otherwise return to Step 2 The improved algorithm (OIPSO) in this paper is the same as PSO, except that Step 1 determines the initial particle swarm via optimization of initial particle swarm.
In the optimization of resource-leveling with fixed duration and the optimization of resources constraints with shortest project duration of a network plan, the following expression determines the initial particle swarm: where [] is related to its start time of work ,  1 [] is the normal duration of the work , rand()% is the random function, and [] is the total float of the work .The limiting conditions are resource variance in the optimization of resource-leveling with fixed duration, as well as resources and project duration in the optimization of resources constraints with shortest project duration.In the Monte Carlo method, using the random function rang()% is the basic principle.
2.2.The Improvement Mechanism of OIPSO.Markov chains are constituted by the PSO M particles [51].And then by randomly selecting initial particles and setting them out to a certain point, the stochastic optimization series of particles constitute the Markov chains.The probability for a particle to set out from  is   , and the probability  ()  for a particle to transfer to  after an  time transfer can be determined by the following formula ( represents the optimal position of a limited number of iterations conditions, it can also be the optimization solution,  ∈ , and  is the state space): where   represents the arrival state of a particle subsequent to  time transfer;  () is called the probability distribution of   ;  (0) = [ 1 ,  2 , . ..] and is the initial distribution which is 1×(+1) matrix of the Markov chains, namely, the probability of the Markov chains starting from ,   = ( 0 = ),  ∈  (state space), and  0 is the state for a particle to set out from;   is equal to the product of  one-time transfer matrix  ( + 1 order phalanx), and  is also called matrix of transition probability, expressed as where   is the probability for a particle to transfer to  in the next time setting out from  [52].
Optimized and unoptimized initial particles comprise the two columns of Markov chains.The probability of particles starting from the location of the initial particle is equal to 100%, while that of particles starting from the locations of other particles is equal to 0. The excellent particle position close to the optimal solution is increased in the n particle positions, which is the position of the excellent initial particle, regardless of the subsequent excellent particle positions that were generated based on the initial particle.There is a higher probability of the excellent particle flying from the position near the optimal solution than it flying away from the optimal solution.Therefore, as shown in (4),  column of one-time  transfer matrix  of Markov chains to optimize initial particle is bigger.
As a result,  column of   and  (0)   of Markov chains to optimize initial particle is bigger.

Solving Large-Scale Network Plans with 223 Works by OIPSO
As shown in Figure 1, a large-scale network plan has a works number of 223 and a calculated project duration of 135.Table 1 shows each work's resources amount, duration, and earliest start time, corresponding to the resource variance of 37.51.The biggest quantity of resources at one period is 27.The optimization of resource-leveling with fixed duration can be unchanged project duration and resource demand equilibrium of each period.The resources supply capacity limit can be met by the optimization of resources constraints with shortest project duration, and it can have minimal extended project duration.

Solving the Optimization of Resource-Leveling with Fixed
Duration.The variance method can be applied to evaluate the resource leveling, and the calculation formula of the variance is where the total number of the samples   is ; the arithmetic average of   is .The evolution equation is [49]   ( + 1) =   () + floor(V  () where  is the number of iterations,   ( + 1) is the dimensional space coordinates of the particle  at ( + 1) times of iterations,   () is the -dimensional space coordinates of the particle  at  times of iterations,  is inertia weight (its general value is 1), V  () is the -dimensional flight velocity of particle ,  1 and  2 are the acceleration constant with a general value of 0-2, rand 1 and rand 2 are the random function with the value in the range of (0, 1),   () is the best position of particle  experienced,   () is the best position of all particles, and floor() is the integral function.corresponding to the resource variance of 22.41.The parameters applied in the OIPSO are the following: the inertia weight  = 1 (empirical value), the acceleration constant  1 = 3.5 (empirical value),  2 = 4.0 (empirical value), the particle number  = 50 (experimental value), the initial particle variance <30 (experimental value), and the number of iterations  = 100 (experimental value).It is obvious that the optimized resource variance (22.41) is much less than the original one (37.51).

Solving the Optimization of Resources Constraints with
Shortest Project Duration.As shown above, the methods of the optimization of resources constraints with shortest project duration in a large-scale network plan and the optimization of resource-leveling with fixed duration are similar.Besides, the initial particle swarm constraint is applied with resources constraints and as small as possible duration.The optimal solutions criteria are different; the constraint range of the feasible solution  The example is as shown in Figure 1, and Table 2 lists the start time of each work for resources constraints with shortest project duration optimization, where 25 is the resource constraint, 199 is the project duration, and 22 is the biggest resources quantity.The applied parameters of the resources constraints with shortest construction period optimization of the OIPSO are the following: the inertia weight  = 1 (empirical value), the acceleration constant  1 = 3.5 (empirical value),  2 = 0.4 (empirical value), the quantity of particle  = 10 (experimental value), the number of iterations  = 1000 (experimental value), the resource constraint of the initial particle swarm is 25 (experimental value), and the constrained duration is 300 (experimental value).The range of and  1 [] is the same as the previous).The resource constraint ( 25) is met, and the corresponding project duration (199) is not too long after optimization.

The Superiority of OIPSO Compared with the Original and the Existing PSOs
After changing optimization parameters, the superiority of OIPSO in optimization of resource-leveling with fixed duration and resources constraints with shortest project duration of a large-scale network plan in Figure 1 is shown in Tables 3  and 4. Case 3 in Table 3 and case 8 in Table 4 are obtained by the original and existing PSOs in which the initial particles are randomly decided.Case 1 in Table 3 and case 5 in Table 4 are obtained by the improved algorithm (OIPSO) in which optimization is used to decide the initial particles.For the optimization of resource-leveling with fixed duration or resources constraints with shortest project duration of a large-scale network, it can be found that the improved algorithm (OIPSO) can accelerate the optimization speed and improve the optimization effect of particle swarm optimization under the same number of iterations by adding proper optimization constraints of the initial particle swarm, such as the variance restriction or resources limitation and project duration constraint.The optimization constraints of the initial particle swarm are decided gradually through the experiment such as case 2 (the resource variance corresponding to the initial particle is 50) and case 4 (the resource variance corresponding to the initial particle is 25) in Table 3 and case 6 (the constrained resources of the initial particle are 25 and there is no constrained project duration of the initial particle), case 7 (there are no constrained resources of the initial particle and the constrained project duration of the   The project duration of the initial particle 135 135 135 135 The resource variance corresponding to the initial particle 30 50 -25 The resource variance corresponding to the optimization results 22.41 22.99 22.99 -"-" denotes "no restrictions" or "no solution in a limited amount of computing time."The other symbols' meanings are the same as the previous.The constrained resources of the initial particle 25 25 --25 The constrained project duration of the initial particle 300 -300 -200 The biggest resources quantity after optimization 22 21 22 19 - The calculated project duration after optimization 199 206 199 206 -Mathematical Problems in Engineering initial particle is 300), and case 9 (the constrained resources and the constrained project duration of the initial particle are, respectively, 25 and 200) in Table 4.

Conclusion
In this paper, the improved particle swarm optimization algorithm via optimization of initial particle swarm (OIPSO) has been proven to improve the solution probability of optimal solution or by the theory of Markov chains in random process and the optimization examples accelerates the optimization speed and improves optimization effect of particle swarm optimization under the same number of iterations.In existing publications on the larger-scale network plan optimization, the optimization examples of the resource-leveling with fixed duration and the resources constraints with shortest project duration on the large-scale network plan with 223 works have the largest work quantity.
[][] + [][] is also different, where the spatial coordinates of particles [][] + [][] are the -dimension variables related to the work start time; their initial values are the coordinates of the initial particle swarm [][]; [][] is the flight speed of particles.

Table 1
also shows the start time of each work for resource-leveling optimization with fixed duration solution,

Table 1 :
The parameters and their optimization solution for the optimization example of the resource-leveling with fixed duration.
"ES" is the early start time of each work."Optimized ES" is the optimized start time.

Table 2 :
The parameters and their optimal solution of resources constraints with shortest project duration.

Table 2 :
Continued.ES" is the early start time of each work."Optimized ES" is the optimized start time. "

Table 3 :
The superiority of OIPSO in optimization of resource-leveling with fixed duration.

Table 4 :
The superiority of OIPSO in optimization of resources constraints with shortest project duration.