Hybrid many-objective particle swarm optimization algorithm for green coal production problem

Information Sciences 518 (2020) 256–271

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Hybrid many-objective particle swarm optimization algorithm for green coal production problem Zhihua Cui a, Jiangjiang Zhang a, Di Wu a, Xingjuan Cai a,∗, Hui Wang b,∗, Wensheng Zhang c, Jinjun Chen d a

Complex System and Computational Intelligence Laboratory, Taiyuan University of Science and Technology, Taiyuan 030024, China School of Information Engineering, Nanchang Institute of Technology, Nanchang 330099, China c State Key Laboratory of Intelligent Control and Management of Complex Systems, Institute of Automation Chinese Academy of Sciences, Beijing 100190, China d Department of Computer Science and Software Engineering, Swinburne University of Technology, Melbourne 3000, Australia b

a r t i c l e

i n f o

Article history: Received 10 August 2019 Revised 10 January 2020 Accepted 10 January 2020 Available online 15 January 2020 Keywords: Coal production Many-objective optimization problems Evolutionary operators Particle swarm optimization (PSO)

a b s t r a c t The key aspect in coal production is realizing safe and efficient mining to maximize the utilization of the resources. A requirement for sustainable economic development is realizing green coal production, which is influenced by factors of coal economic, energy, ecological, coal gangue economic and social benefits. To balance these factors, this paper proposes a many-objective optimization model with five objectives for green coal production. Furthermore, a hybrid many-objective particle swarm optimization (HMaPSO) algorithm is designed to solve the established model. A new offspring of the alternative pool is generated by employing different evolutionary operators. The environmental selection mechanism is adopted to select and store the excellent solutions. Two sets of experiments are performed to verify the effectiveness of the proposed approach: First, the HMaPSO algorithm is tested on the DTLZ functions, and its performance is compared with that of several widely used many-objective algorithms. Second, the HMaPSO algorithm is applied to solve the many-objective green coal production optimization model. The computational results demonstrate the effectiveness of the proposed approach, and the simulation results prove that the designed approach can provide promising choices for decision makers in regional planning. © 2020 Elsevier Inc. All rights reserved.

1. Introduction Coal resources play a significant role as energy sources and industrial raw materials. The energy pattern in China involves inferior oil and gas resources, but rich coal resources, implying that coal is expected to be the cornerstone of energy security and stable supply for a long time in China. However, the economy of the coal industry has encountered severe challenges in recent years, owing to the decreasing demand for coal with the modification in the economy policy and the increasing proportion of non-fossil energy owing to the energy structure optimization [1,2]. Therefore, promoting green coal production has become a critical requirement to achieve sustainable economic development.

∗

Corresponding authors. E-mail addresses: [email protected] (X. Cai), [email protected] (H. Wang).

https://doi.org/10.1016/j.ins.2020.01.018 0020-0255/© 2020 Elsevier Inc. All rights reserved.

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In general, although the economic benefits in the process of coal production receive considerable attention, the other factors, including the increasing energy input, overload capacity of environment, and continuing social negative impact are usually ignored. Therefore, the development of an efficient coal production optimization method is of significance for realizing the sustainable utilization of the coal resource system and achieving economic development. To adapt to the development requirements, several researchers formulated various coal development programs. In addition, several excellent methods to solve practical problems were developed [3,4]. Menget et al. [5] successfully solved the reservoir problem by improving the correlation model. Che and Jia [6] built a three-dimensional geological model of coal seams by using the weighted kriging method and multi-source data. Jiang et al. [7], taking a coal mining machine as a case study, described a conceptual scheme improvement approach based on the performance value of the principle solution. Jiang et al. [8] designed the moving-object tracking algorithm based on the principal component analysis-scale invariant feature transform (PCA-SIFT) to optimize the underground coal mines. Safdarnejad et al. [9] developed a dynamic model for a coal-fired utility boiler to forecast and minimize NOx and CO emissions simultaneously. Yang et al. [10] performed the vibration test of a single coal gangue particle directly impacting the metal plate and performed recognition based on the vibration signal and stacking integration. In addition, Wu et al. [11] proposed a method to evaluate the gas production from multiple coal seams from an economic perspective. However, the above methods explored the coal development from only a single perspective and did not comprehensively consider the coal development requirements. This work was aimed to address the abovementioned limitation, and the main contributions are as follows: (1) An efficient many-objective optimization model for green coal production is designed to help decision makers gain good references in regional planning. In this approach, five optimization objectives including the coal economic, energy, ecological, coal gangue economic, and social benefits are simultaneously considered. (2) To solve this model, a hybrid many-objective particle swarm optimization (HMaPSO) algorithm is proposed, whose complexity is the same as that of other many-objective evolutionary algorithms. (3) Two sets of simulation experiments on the DTLZ benchmark set and green coal production problem are conducted to evaluate the performance of the HMaPSO. The remaining paper is organized as follows. The formulation of the many-objective optimization model for green coal production is described in Section 2. The proposed hybrid many-objective PSO is explained in Section 3. The performed validation tests are described in Section 4, and the conclusions are derived in Section 5. 2. Green coal production problem 2.1. Problem description Coal is the foundational energy of national economy development in China. Consequently, optimizing the coal production is significant to realize energy sustainable development. Coal production refers to all the production activities in which people exploit underground resources. In general, it includes three processes, namely, basic production, auxiliary production, and service production. Basic production involves activities that directly correspond to the extraction of useful minerals, such as underground development, tunneling, mining, and underground transportation lifting. Two types of coal mining processes may be employed: open-pit and underground. According to the current situation of the coal industry, this study focuses on the underground mining for coal production. The steps pertaining to this type of coal production process are shown in Fig. 1, which indicates that the underground mining process involves the tunneling, recovery and transportation lifting processes. Coal processing involves coal preparation and coal washing, and the produced coal is used for export and power generation, among other purposes. Any increase in the coal production inevitably leads to the higher generation of pollutants and higher energy consumption. Furthermore, if the environmental and energy benefits are insufficient, the economic benefits of the coal production

Coal Production Underground

Tunneling

Recovery

Coal processing Transportation lifting

Coal preparation

Coal washing

Raw coal transportation Coal power generation Other purposes

Energy consumption: Electricity, Coal

Pollutant emission: SO 2, Mine water, Coal gangue Fig. 1. Process flow of coal production.

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in the enterprises may be reduced, along with the social benefits of the enterprises. In other words, the economic, environmental, energy benefits and social benefits of coal enterprises are closely related; therefore, establishing a good balance among these factors is important to realize green coal production. Because several factors need to be comprehensively considered in a green coal production process, this problem is a complex multi-objective optimization [12,13] or many-objective optimization problem [14,15]. 2.2. Many-objective optimization model Several scholars developed different optimization models for the coal production [16]. Zheng [17] discussed the optimal management mode of China’s regional economy by combining multi-objective planning with the basic and dynamic equilibrium models in the input-output method. Cui et al. [18] adopted a multi-objective planning model considering the maximum gross domestic product (GDP) and maximum non-coal industry GDP with the minimum positive and negative deviations of the dynamic input-output of various industries. Yu [19] built a goal programming model to minimize the energy consumption and maximize the GDP of economic growth. According to the development plan of the region in a few years, Narisa [20] presented a multi-objective dynamic input-output optimization model, in which the cumulative value of GDP, proportion of the added value of the tertiary industry, average speed of the GDP, lowest energy consumption, and lowest pollution level were considered. Yang [21] constructed a multi-objective optimization model considering three objectives: coal output, total energy consumption, and main pollutant discharge. In this work, a many-objective optimization model was designed for green coal production. In contrast to the existing methods, this model also takes into account the factors of coal economic, energy, ecological, coal gangue economic and social benefits. Moreover, this model involves multi-variable optimization. To solve the multi-variable optimization problems, some excellent methods have been proposed. Chakrabortty et al. [22] designed a multi-mode resource-constrained project scheduling strategy by using the modified variable neighborhood search heuristic approach. Kose and Tasci [23] investigated the geodetic deformation forecasting based on the multi-variable gray prediction model and regression model. Alaniet and Osunmakinde [24] solved the multiple weather variable matter in the intelligent short term load forecasting of the electricity demand for smart homes. Blanqueroet et al. [25] carefully analyzed the variable selection mechanism for the classification of multivariate functional data. Li et al. [26] proposed multi-variable regression methods by using the class 2 modified Chebyshev polynomials. 2.2.1. Objective function Five objective functions are considered for the many-objective green coal production process, namely, the coal economic, energy, environment, coal gangue economic, and social benefits. These objective functions are defined as follows. (1) Coal economic benefit (f1 ) The coal economic benefit is an important objective, which considerably affects the coal production. In general, producing more coal lead to higher economic benefits. The coal production process involves four key operations, namely, tunneling, recovery, underground transportation lifting, and coal preparation. Tunneling provides mining space for the formal coal mining. The recovery process helps excavate the underground raw coal. Underground transportation lifting refers to the transportation of coal from the ground to the surface. Coal preparation involves removing the excess impurities in the coal obtained after the lifting process to produce the final available coal. The coal economic benefit is generated by selling the coal after the preparation process. However, the price of coal is usually influenced by the market-oriented supply and demand factors, which are difficult to represent quantitatively. Therefore, the coal economic benefit can be calculated by considering the cumulative production of coal after the coal preparation process during the planning period, as follows:

f1 = max

T 

Y iel dt,4

(1)

t=1

where T is the number of coal planning years; and Yieldt ,4 denotes the coal yield of the fourth key process in the t-th year. (1) Energy benefit (f2 ) Energy is critical for promoting economic development. However, the cost of energy exploitation is growing rapidly, and the corresponding production is gradually decreasing yearly. To relieve the stress of the cost, the utilization of energy needs to be improved and consumption of energy needs to be reduced. Therefore, the energy benefit can be defined by evaluating the energy consumption, which can be reduced by updating the transition. The objective function can be calculated as

f2 = min

T  S  I 

(At,s,iY iel dt,s − Bt,s,iYt,sY iel dt,s )

(2)

t=1 s=1 i=1

where S is the key process number; I denotes the types of energy consumption; Yieldt , s represents the coal yield in the sth key process in the tth year; At , s , I is the unit consumption of the ith energy in the sth key process in the tth year; Bt , s , I is the unit frugal consumption of the ith energy in the sth key process in the tth year after updating the transition; and Yt , s indicates whether the sth key process is an updated transition. Yt , s =0 denotes that the sth key process is not an updated transition, whereas Yt , s =1 means that the sth key process is an updated transition.

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(1) Environmental benefit (f3 ) The environmental benefit is an important objective for green coal production as coal production leads to several environmental problems such as the pollution of wastewater and generation of solid waste, sulfur dioxide, and dust. The pollutant emissions can be reduced through treatment and utilization. Thus, the objective function of environmental benefit can be defined as

f3 = min

T  S  L 

(Ct,s,l Y iel dt,s − (1 −

t=1 s=1 l=1

K 

(αt,k,l Zt,k ) ))

(3)

k=1

where L denotes the types of pollutants in the coal production; K denotes the types of pollutants in the planning treatment; Ct , s , l is the unit pollutant discharge of the lth pollutant in the sth key process in the tth year;α t, k, l is the utilization rate of the lth pollutant in the kth pollutant in the tth year; Zt , k indicates whether the kth pollutant is managed in the tth year. Zt , k =0 denotes that the kth pollutant is not managed, whereas Zt , k =1 indicates that the kth pollutant is managed. (1) Coal gangue economic benefit (f4 ) With the societal development, many production industries have attempted to utilize waste effectively. For example, the coal gangue can be transformed to a hollow gangue brick, which can be used as an energy-saving wall material and for power generation. In this manner, the coal gangue can generate economic benefits. The objective function of the coal gangue economic benefit can thus be defined as

f4 = max

T 

GanY iel dt

(4)

t

where GanYieldt denotes the coal gangue yield in the tth year. (1) Social benefit (f5 ) The social benefit refers to the safety in the green coal production. To reduce safety losses, enterprises invest certain safety resources. However, the safety production requirements may not be met if the input cost is insufficient. Therefore, ensuring a reasonable safety input cost is important for green coal production. The objective function of the social benefit can be defined as

f5 = max

T H  

(

t=1

Dt,hY iel dt,4 + g(t ))

(5)

h=1

where H is the number of safety input projects; Dh , t denotes the investment funds for the unit coal production in the hth safety input project in the tth year; and g(t) denotes the accident loss function. In general, the accident loss function can be expressed by the C-D production function, the general representation of which is as follows.

g(t ) = R(Mt,1Y iel dt,4 )β1 (Mt,2Y iel dt,4 )β2 (Mt,3Y iel dt,4 )β3 (Mt,4Y iel dt,4 )β4 (Mt,5Y iel dt,4 )β5

(6)

where R is the level coefficient of technological development; Mt ,1 , Mt ,2 , Mt ,3 , Mt ,4 , and Mt ,5 are the input costs of labor protection, industrial hygiene, publicity education, safety technology, and auxiliary facilities for the unit coal production in the tth year, respectively; β 1 , β 2 , β 3 , β 4 , and β 5 are the output elasticity coefficients corresponding to the different safety inputs. 2.2.2. Constraint handling To match the actual coal production, some constraints are required to be satisfied [27,28]. In general, the tunneling process is a production activity performed in advance to realize the normal mining of coal in the mining process. In addition, the underground transportation and lifting of coal are based on the coal obtained from the excavation and mining. Therefore, the following constraint can be defined.

Y iel dt,3 < Y iel dt,1 + Y iel dt,2

(7)

Coal preparation can be achieved by upgrading the process or by obtaining coal as raw coal. However, raw coal involves several impurities, which need to be treated through the coal preparation process to improve the coal quality. Thus,

Y iel dt,3 > Y iel dt,4

(8)

Most of the coal resources cannot be exploited owing to the limitations of mining technology and cost. T  t=1

(Y iel dt,1 + Y iel dt,2 ) R

< RES

where R is the recovery rate, and RES is the amount of total coal resources.

(9)

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The sum of various safety investment funds should be more than the average annual safety investment cost. H 

Dt,h Xt,4 ≥ CostLimit1

(10)

h=1

whereCostLimit1 denotes the lower limit for the preventive safety investment. A portion of the funds for the energy conservation and emission reduction should help meet the production requirements. However, it is unrealistic for an enterprise to devote all its income to energy-saving renovation and emission control. The investment funds of the energy and environmental efficiency should thus generally be less than the revenue costs of the enterprises.



T S   t=1

IYt,s · Yt,s +

s=1

K  

IZt,k · Zt,k

 

≤ CostLimit2

(11)

k=1

where CostLimit2 is the amount of allowable funds for the energy conservation and emission reduction; IYt, s represents the investment funds required for updating the transition in the sth key process in the tth year;IZt, k represents the investment funds required for managing the pollutants in the kth project in the tth year. The maximum utilization rate of the various pollution treatment or comprehensive treatment can only reach 1, and thus it is necessary to restrict the utilization rate to K  less than 1. That is, (αt,k,l Zt,k ) ≤ 1. k=1

To overcome the above constraints, the modified constraint domination principle is adopted. The value of the constraint violation needs to be calculated before solving the problem. Based on the value of the function and constraint violation, the dominance relationship among different individuals can be obtained. 3. Proposed approach for green coal production In the past decade, many scholars have proposed various excellent algorithms to solve the many-objective optimization problems (MaOPs) [29–31], such as evolutionary algorithm based on selection-and-elimination [32], stochastic ranking mechanism based on multiple indicators [33,34], hybrid many-objective cuckoo search algorithm (HMaOCS) [35], balanceable fitness estimation based on particle swarm optimization (NMPSO) [36] and bi-criterion evolutionary algorithm. These algorithms aim to solve the balance problem in terms of the convergence and diversity [37]. Compared to the other algorithms, the NMPSO can better balance the diversity and convergence because of the use of the novel fitness selection mechanism. To further improve the performance of the NMPSO, an enhanced NMPSO variant is designed. The details of the proposed approach are described later in the section. 3.1. Original NMPSO In the original NMPSO, a new offspring is generated using the particle swarm optimization (PSO) algorithm. To select an excellent offspring, an environmental selection mechanism is employed. The PSO operator and environment selection operator in the NMPSO are described in this subsection. (1) PSO operator PSO is an excellent swarm intelligence algorithm, and several improved algorithms based on the PSO have been proposed [38,39]. To guide the better evolution of the entire population in the MaOPs, the improved velocity update equation is used, as described in the original literature. In addition, under the influence of the environmental selection operator, an archive is established, which contains the optimal solution of each iteration. The following relations are used to update the velocity and position.

velocit yi (t + 1 ) = wvelocit yi (t ) + c1 r1 ( position pbesti − positioni (t )) + c2 r2 ( positiongbesti − positioni (t )) + c3 r3 ( positiongbesti − position pbesti ) positioni (t + 1 ) = positioni (t ) + velocit yi (t + 1 )

(12)

where t is the iteration number; w is the inertial value; c1 , c2 and c3 are three learning factors; r1 , r2 and r3 are three uniformly distributed random numbers in [0,1]; and position pbesti and positiongbesti are the positions of the local-best and the global-best particle, respectively. These values are randomly chosen from the current evolutionary population and the top 10% of the archive by using the evaluation mechanism in a descending order, respectively. The updating strategy can provide another evolutionary direction from the local-best particle pointing to the global-best individual. In addition, this approach can improve the convergence of solutions on MaOPs. (1) Environmental selection operator The solutions are generated using different evolutionary operators, and the ones with a better quality in terms of the convergence and diversity are selected on the basis of the environmental selection mechanism. An archive based on the balanceable fitness estimation (BFE) approach is established to store the best solutions in the course of the population

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Algorithm 1 Environment Selection (A, V) Operator. Begin for i=1 to |V| for j=1 to |A| Judge the dominance relation between Ai and Vi ; if the nondominated solution is located in A Retain the corresponding nondominated Ai solutions; if the nondominated solution lies in V Add the corresponding nondominated Vi to A; end if end if end for if |A| > |N| Compute the fitness values using the BFE method; Remove some solutions with the worst fitness values; end if end for Output the archive A; End

evolution [36]. In addition, the BFE method has been proved to be effective for overcoming the problem of both the Paretoranking and decomposition methods. The distance in the convergence and diversity are considered in the BFE method to balance these two issues in the objective space. Assume that the population P = { p1 , p2 , ..., pN }contains N individuals, the BFE method can be expressed as follows.

f itness( pi , P ) = α ∗ Discon( pi , P ) + β ∗ Disdiv( pi , P )

(13)

whereDiscon(pi , P)andDisdiv(pi , P)respectively denote the distance ofpi in terms of the diversity and convergence;α andβ are the adjustment factors used to tune the influence of these two distances via dynamic balancing, respectively. In other words, the factor values adaptively change according to the different individuals of the two distances. The principle can be found in [36]. The largest and smallest values of the corresponding objective should be used for normalization when computing the BFE value. Such normalization reduces the float range on many objectives. To prevent the obtained archive from exceeding the population size boundary, a truncation selection mechanism is used, which can effectively guide the search direction to approach the true PF. In the process of environmental selection, the dominance relation between the archive A and the new generated population V are first verified. The nondominated solution is added to the archive if the solution in A non-dominates the new solution in V. Conversely, the dominated solution is removed from A. The truncation mechanism is executed when the size of A exceeds the population size. The worst solution is deleted until the size of A meets the specified population size according to the BFE value. The main steps of the Environment_Selection (A, V) operator are listed in Algorithm 1, where A is the archive, and V is the offspring population. (1) Discussion and analyses Although the environmental selection operator in the NMPSO achieves a satisfactory performance in balancing the solutions of convergence and diversity, the new offspring generated by the PSO operator may reduce the diversity of the alternative solutions. To tackle this problem, the idea of integration is introduced in the proposed approach. By integrating many excellent evolutionary operators to construct the alternative pool, the algorithm, under the selection pressure, ensures that the operators coordinate with each other to obtain more extensive alternative solutions in the evolutionary process of the population. Therefore, the overall evolutionary efficiency of the algorithm is improved owing to the effect of the alternative pool. The convergence and distribution of the population is ensured owing to the influence of the environmental operator. 3.2. Construction of alternative pool To construct the alternative pool, many excellent evolutionary operators are employed, and the new offspring is generated by the different evolutionary operators. In the proposed approach, the differential evolutionary (DE) and simulate binary crossover (SBX) operators are introduced and coordinated with the PSO operator to construct the alternative pool. (1) DE operator The basic idea of a DE is to use the diverse vectors of two individuals randomly chosen from the population to act as mutant individuals [40,41]. The offspring is generated by the recombination of the mutant and current individuals. The

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Q1˖Generated by DE

Q12

Q2˖Generated by PSO Q3˖Generated by SBX

Q2

Q1

Q123 Q13

Q23

Q3

Fig. 2. Alternative pool of solutions generated by different evolutionary operators.

excellent individual is obtained by comparing the fitness values of the two individuals. As a result, the population can be guided to evolve in a better direction. However, the original DE operator cannot balance the solutions of convergence and diversity when solving the MaOPs. To effectively guide the entire population to solve the MaOPs, the original DE operator is improved as follows.



Posi + F ∗ (Pos j − Posi ) + F ∗ (Posk − Posi ) i f r and ≤ CR Pos j or Posk otherwise

Posi =

(14)

where F is the contraction factor; Posi is randomly chosen from the current population; Posj and Posk are randomly chosen from the top 10% individuals in the archive; rand is a random number within [0,1]; and CR is the crossover probability. (1) SBX operator The SBX operator has been proved to be effective in solving the MaOPs [42]. This operator imitates the human reproduction approach based on a binary string and can protect the relevant pattern information of the parent chromosomes in the offspring. The parents Xj 1 and Xj 2 are randomly chosen from the evolutionary population, and two offspring Cj 1 and Cj 2 are generated, as follows.









C 1j = 0.5 (w j + v j ) − β1 (w j − v j ) , C 2j = 0.5 (w j + v j ) − β2 (w j − v j ) ,

(15)

where j=1,2,...,n is the jth component, wj and vj denote the largest and smallest values of eterβk (k = 1, 2 )is defined as



βk =

1

[rk .ak ] η+1

1

[1/(2 − rk .ak )] η+1

i f rk ≤ 1/ak otherwise

,

X j1 andX j2 ,

respectively. The param-

(16)

whererk (k = 1, 2 )are random numbers within [0, 1]; ηis the crossover distribution index; and ak (k = 1, 2 )are defined as follows.



ak =



−(η+1) k=1

−(η+1) 2 − 1 + 2 ( u j − w j )/ ( w j − v j ) k=2 2 − 1 + 2 ( v j − l j )/ ( w j − v j )

(17)

wherewj = vj ; and lj and uj are the two bounds of the j-th decision variable, respectively. Next, an integer k is randomly generated to decide which one of the offspring solutions (Cj 1 and Cj 2 ) is further permuted by the polynomial-based mutation (PM). The PM is executed on Ck , and the new objectives are evaluated using the evaluation mechanism. In this manner, the excellent solutions on each generation can be recorded, and each offspring is added into the offspring set. To consider the influencing factors of solving the MaOPs, the original SBX operator needs to be modified. For the two parent individuals, Xj 1 is randomly selected from the current evolutionary population, and Xj 2 is similar to the selection rule of positiongbesti in the PSO operator. To better understand and visualize the idea of the ensemble evolutionary operator, a schematic regarding the alternative pool is expressed in Fig. 2. It can be seen that Q1 , Q2 , and Q3 are the offspring generated by the DE, PSO and SBX operators, respectively. Some particular cases are described by the overlapping area of Fig. 2. In some cases, the same offspring may be

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Algorithm 2 Framework of the HMaPSO. Begin Initialize the populations P and the related parameters; Initialize archive A; while gen
Table 1 Coal production in a certain coal enterprise (tons). Year

2011

2012

2013

2014

2015

Coal Yield

1808.10

2002.11

1917.90

1883.33

1863.65

generated by different evolutionary operators, which cannot be ignored in the process of population evolution even if the occurrence probability is small. For example, Q12 denotes the same offspring produced by the DE and PSO operators; Q13 and Q23 represent the same offspring generated by the DE and SBX operators, and the PSO and SBX operators, respectively. Q123 is the same offspring generated by the DE, PSO and SBX operators. Notably, the alternative pool (Q=Q1 +Q2 +Q3 ) provides more excellent alternative solutions than the offspring generated by the PSO operator (Q2 ) for environmental selection. 3.3. Framework of the HMaPSO In the proposed approach, HMaPSO, an initial populations P is randomly produced. The initial archive A is filled with the individuals selected from the first rank of populations P based on the nondominated sorting method. Next, an alternative pool is constructed by integrating the three evolutionary operators, namely, the DE, PSO and SBX operators. The new offspring in the population Q are generated by the three evolutionary operators in the alternative pool. To select the excellent offspring generated by the different evolutionary operators in the alternative pool, the environmental selection mechanism is employed. The framework of the proposed HMaPSO is described in Algorithm 2, where Q1 , Q2 , Q3 , and A∗ denote the offspring generated by the different evolutionary operators; gen is the generation index, and Maxgen is the largest number of generations. 3.4. Complexity analysis In the proposed approach, the stop condition is set to the maximum number of generations. The main environmental selection mechanism is the same as that of the original NMPSO, and the only difference is in the construction of the alternative pool. The most time-consuming evolutionary operator in terms of the complexity in the alternative pool is the environmental selection operator. The maximum time consumption in the case of the environmental selection operator occurs mainly when performing the nondominated ranking mechanism or comparing the fitness values. In particular, the maximum population size in the alternative pool can reach MN, (Q=Q1 ∪Q2 ∪Q3 ). After performing the nondominated ranking mechanism or the comparison of fitness values, the computational complexity is O(MN2 ). It is obvious that the time complexity of the environmental selection operator is O(MN2 ). In addition, the DE, PSO, and SBX operators, which are used for generating the offspring, have a similar time consumption of O(MN2 ) in the alternative pool. It can be noted that the maximum iteration time, as a fixed constant, is considerably smaller than the number of nondominated solutions. Hence, the time complexity of our approach is O(MN2 ). 4. Simulation experiments 4.1. Correlation parameter (1) Parameter settings for the model The many-objective optimization model for green coal production involves several parameters. Table 1 lists the coal production amounts per year in a certain coal enterprise. The parameter values in the equation were obtained by em-

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Z. Cui, J. Zhang and D. Wu et al. / Information Sciences 518 (2020) 256–271 Table 2 Relevant parameters of the coal model. Parameter

Value

Total number of years Number of key processes Types of energy consumption Types of pollutants in coal production Types of pollutants of planning management Number of safety input projects

T=5 S=4 I=2 L=2 K=2 H=5

Table 3 Energy consumption per unit output of key processes. Key process

Raw coal (ton/ ton)

Electricity (kWh/ ton)

Tunneling Recovery Underground transportation lifting Coal preparation

0.0025 0.0025 0.0025 0.0025

8.79 6.38 5.01 10.06

Table 4 Unit energy consumption of key processes after energy conservation reform. Key process

Raw coal (e−2 ton/ ton)

Electricity (kWh/ ton)

Tunneling Recovery Underground transportation lifting Coal preparation

0.014 0.023 0.026 0.037

0.971 0.750 0.297 1.006

Table 5 Emissions of pollutants per unit output of key processes. Key process

SO2 (e−4 ton/ ton)

Mine water (ton/ ton)

Coal gangue (ton/ ton)

Tunneling Recovery Underground transportation lifting Coal preparation

0.4937 0.4849 0.4667 0.4487

0.0631 0.0543 0 0

0.4839 0.2803 0 0

Table 6 Emission reduction rate and utilization rate for comprehensive treatment of emissions (All values in percent points). Emissions

2016

2017

2018

2019

2020

SO2 Mine water Coal gangue

16.70 82.20 30.00

15.60 83.00 33.00

44.10 82.10 37.90

42.10 82.30 35.40

40.00 83.70 79.00

ploying the multivariate linear regression method and SPSS. To gain a convincing result, the Friedman test was performed to demonstrate the parameter accuracy of the loss function with a significance level of 0.05. Therefore, the parameters involved in the loss function were determined from the above analysis. The parameters were set as R=1275.38, β1 = − 0.07,β2 = − 0.084,β3 = − 0.459,β4 = − 0.042, andβ5 = − 0.163, and the values of the other parameters are listed in Table 2. Coal is primarily consumed for power generation and heating. Therefore, the proportion of coal consumption in the production process can be equally divided into four key energy consumption processes. Table 3 presents the unit output of the key processes in coal mining enterprises and the consumption of raw coal, according to the actual energy consumption in 2015. By comparing the productivity of the new and old equipment units, it can be noted that the energy consumption and saving rate is approximately 0.1%. Therefore, the unit energy consumption of the key processes after the energy conservation reform is given in Table 4. In addition, the emissions of the pollutants per unit output of the key processes is presented in Table 5. Based on the energy consumption and emissions of the coal production in 2015, the coal production in 2016–2020 can be simulated. According to the development plan of the coal enterprises, the emission reduction rate and utilization rate of the comprehensive emission control for 2016–2020 are presented in Table 6. The cost of the energy-saving renovation of the key processes and treatment or comprehensive utilization of the pollutant discharge for 2016–2020 is presented in Table 7. (1) Parameter settings for the involved algorithms

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265

Table 7 Costs of energy-saving renovation of key processes and treatment or comprehensive utilization of pollutant discharges (Billion Yuan). Investment project

2016

2017

2018

2019

2020

Energy-saving renovation Comprehensive management and utilization

2.7800 5.2000

2.4900 2.6800

2.2000 2.7100

6.8382 0.7000

2.00891 1.2600

The HMaPSO is adopted to solve the many-objective green coal production problem. To examine the influence of the proposed approach, the HMaPSO is compared with several widely used algorithms including the NSGA-III, REVA, MOEA/D-DE, KnEA, and PICEA-g algorithms. To obtain convincing results, all the key parameters are set as defined in the original literature. As mentioned previously, two series of experiments are performed: (1) testing on the DTLZ benchmark set [43]; and (2) solving the many-objective green coal production problem. For problems with different numbers of objective, different population sizes are employed. When the number of objectives is 3, 5, 8, 10 and 15, the population size is set as 91, 210, 156, 275 and 135, respectively. The two-layer reference point mechanism is adopted in the NSGA-III, RVEA and MOEA/D-DE algorithms. The probability of the SBX and PM are set as 1 and 1/D, respectively. The largest number of generations (Maxgen) is the stopping condition for all the algorithms. In addition, each algorithm is tested 20 times on all the functions. 4.2. Performance metrics In the experiments, four performance metrics are considered to evaluate the influence of the algorithms. These metrics can reflect the influence of the algorithm and are extensively adopted in solving the MaOPs [44]. (1) Inverse generation distance (IGD) The IGD has been extensively adopted to comprehensively evaluate the influence of the algorithm on the distribution and convergence for a benchmark problem [45], and it can be described as



IGD =

n  i=1

di2 (18)

PF ∗

where n is the number of solutions in the true PF∗ , and di represents the Euclidean distance from the solution i of the PF∗ to the solution of the approximated PF. A smaller IGD corresponds to a better performance. (1) Hypervolume (HV) The HV is used to represent the volume covered by the obtained PF in the object region, defined as the HV between the front surface and the reference vectorr = (r1 , r2 , ..., rm )T [46]. Therefore, the HV metric reflects the solution distribution of the PF and can be defined as follows:

HV (x, r ) = V OL(

N

[ f1 (x ), r1 ] × ...[ fm (x ), rm ])

(19)

i=1

where VOL() denotes the Lebesgue measure, and N is the number of the obtained PF. A higher HV corresponds to a better performance. (1) Spacing (SP) The indicator spacing (SP) is employed to measure the distribution of the nondominated solution sets [47] and can be expressed as follows:



SP =

1 Nnonsol



Nnonsol

2

( di − d )

(20)

i=1

whereNnonsol is the number of nondominated solutions; di is the Euclidean distance between the nondominated solution i and the nearest solution; and dis the the average value of di . The nondominated solutions are equally distributed when SP=0. (1) Coverage The coverage metric is often used to measure the coverage of the Pareto solutions by evaluating the proportion of the dominated solutions, in other words, by determining how many more Pareto solutions can be searched by the evolutionary algorithms in the whole solution space [48].

coverage(P Fx , P Fy ) =

|{b ∈ PFy |∃a ∈ PFx : a b}| |PFy |

(21)

where PFx and PFy are two Pareto optimal sets, respectively. When the value of coverage(P Fx , P Fy )=1, all the solutions inPFx are dominated individuals. Otherwise (i.e.,coverage(P Fx , P Fy )=0), all the solutions in PFy are nondominated individuals.

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4.3. Results on the DTLZ Benchmark Set 4.3.1. Comparison of HMaPSO with five other algorithms Table 8 presents the IGD values obtained using the HMaPSO, NSGA-III, RVEA, KnEA, MOEA/D-DE, and PICEA-g on the DTLZ benchmark. The best result on the different test instances is highlighted. Based on a Wilcoxon’s rank sum statistical test, the labels ‘+,’ ‘-,’ and ‘=’ show that the results acquired by the different algorithms are significantly superior, worse, or equal to those obtained by the proposed approach, respectively. The results indicate that the HMaPSO obtains the best performance on 27 out of 35 functions. The RVEA performs better than the other algorithms on three test instances. Both the MOEA/D-DE and PICEA-g achieve the best results on two functions. NSGA-III achieves the best result on only one test instance. However, the KnEA fails to obtain the best result on any of the 35 test instances. The results demonstrate that the proposed HMaPSO exhibits a relatively reliable performance on the DTLZ benchmark set. The HMaPSO achieves a performance similar to that of the KnEA and MOEA/D-DE on DTLZ1 with 3, 10, and 15 objectives. For the other two test instances of the DTLZ1, the HMaPSO exhibits a better performance than that of the other five algorithms. The MOEA/D-DE exhibits a better performance than that of the HMaPSO on DTLZ3 with three objectives. On this instance, the RVEA, KnEA, PICEA-g, and HMaPSO obtain similar performance. For the other test instances, the HMaPSO is better than, or at least similar to the five other algorithms. The new offspring is generated by a single evolutionary operator in the alternative pool. The Pareto-ranking or decomposition method are adopted to select better solutions and guide the population to evolve in a relatively better direction. However, this aspect cannot lead to the sufficient selection pressure to approach the true PF. In the proposed HMaPSO, the alternative solution of the population generated by the different evolutionary operators can overcome the restrictions of the Pareto-ranking and decomposition approach. Therefore, the HMaPSO can demonstrate superior performance on the DTLZ test functions. 4.3.2. Performance analysis of the alternative pool Three evolutionary operators are introduced to build the alternative pool in the HMaPSO, which is supposed to strengthen the performance of the NMPSO. To measure the influence of the proposed alternative pool, the HMaPSO is compared with the original NMPSO. Table 9 lists the IGD values achieved using the HMaPSO and NMPSO on the DTLZ test instances. The HMaPSO outperforms the original NMPSO on most instances. Among the 35 test instances, the HMaPSO achieves better results on 25 instances, and NMPSO exhibits a better performance on the remaining 10 instances. For DTLZ1–DTLZ3, the HMaPSO is significantly better than the NMPSO on most instances. The HMaPSO exhibits a slightly worse performance than that of the NMPSO on DTLZ6 with 3 and 15 objectives, likely because the obtained Pareto front of HMaPSO is similar to that of the original NMPSO. The effectiveness of the HMaPSO is not affected when solving the other MaOPs. It is noted that the worst performance of the proposed method is similar to that of the original NMPSO on the remaining test instances. In general, the HMaPSO is clearly superior to the original NMPSO. Therefore, it can be concluded that the concept of the introduced alternative pool can significantly enhance the performance of the algorithm. 4.4. Results on the Green Coal Production Problem As described in Section 4.3, the proposed HMaPSO was tested on the DTLZ, and the results indicated that the HMaPSO exhibits a superior performance and outperforms the five other widely used algorithms. As described in this section, the HMaPSO was employed to solve the many-objective optimization model for green coal production. Similar to in Section 4.3, the HMaPSO was compared with the NSGA-III, RVEA, KnEA, MOEA/D-DE, and PICEA-g algorithms according to different metrics indicators, such as the HV, spacing and coverage. However, the true Pareto front was required to allow the use these measures. To this end, the PF resulting from the union of the PFs of all the methods was considered as the true PF [49]. Fig. 3 presents the result obtained using the different algorithms. The conflicts between the objectives can be clearly demonstrated. It is concluded that the result of the HMaPSO is better than those of the RVEA, KnEA, and MOEA/D-DE. A similar effect on the coal model is obtained when using the NSGA-III and PICEA-g. However, the NSGA-III and PICEA-g have proved their effectiveness on many application issues. Therefore, the effectiveness and higher accuracy of HMaPSO can be proved in solving the coal production problem. For further statistical analysis, the comparison results based on the HV, SP and coverage are considered, as obtained using the Wilcoxon’s rank sum test. Table 10 indicates that the HV values corresponding to the HMaPSO and other algorithms do not exhibit a significant difference. However, a significant difference can be noted in terms of the coverage. This aspect indicates that the HMaPSO covers all the true PF. In addition, the NSGA-III, RVEA, KnEA, and PICEA-g achieve a similar performance in terms of the coverage when solving the coal model, likely because all three algorithms use the reference point strategy. In terms of the SP, RVEA ranks first, and the five other algorithms can be ranked as follows: KnEA, PICEA-g, NSGA-III, HMaPSO, and MOEA/D-DE. The main reason is that the reference point is vital for the distribute selective of the solutions. The above analysis demonstrates that the proposed many-objective optimization model for green coal production is reasonable. Moreover, the proposed HMaPSO demonstrates a promising performance and outperforms the other algorithms on

Table 8 Comparison results of different algorithms on the DTLZ for IGD values. M

NSGA-III

RVEA

KnEA

MOEA/D-DE

PICEA-g

HMaPSO

DTLZ1

3 5 8 10 15 3 5 8 10 15 3 5 8 10 15 3 5 8 10 15 3 5 8 10 15 3 5 8 10 15 3 5 8 10

2.2336e+1 (4.60e+0) 2.5708e+1 (3.42e+0) 2.5146e+1 (6.28e+0) 2.7640e+1 (6.04e+0) 2.3966e+1 (5.81e+0) 4.3371e-1 (3.23e-2) 6.0515e-1 (2.59e-2) 9.0694e-1 (2.97e-2) 1.0026e+0 (3.00e-2) 1.2109e+0 (3.12e-2) 2.2727e+2 (3.06e+1) 3.4149e+2 (5.91e+1) 3.2421e+2 (6.23e+1) 3.7223e+2 (5.47e+1) 3.1391e+2 (7.94e+1) 8.2796e-1 (1.21e-1) 9.4744e-1 (6.19e-2) 1.0889e+0 (6.61e-2) 1.1369e+0 (5.77e-2) = 1.2352e+0 (5.80e-2) 3.6398e-1 (4.50e-2) 3.4688e-1 (3.26e-2) 3.5943e-1 (3.80e-2) 3.7118e-1 (4.32e-2) = 3.8071e-1 (5.81e-2) 8.0490e+0 (2.43e-1) 8.1122e+0 (1.48e-1) 8.1832e+0 (1.91e-1) 8.3808e+0 (7.01e-2) = 8.3233e+0 (3.08e-1) 8.2998e+0 (6.66e-1) 1.3736e+1 (1.01e+0) 2.3298e+1 (1.93e+0) 2.9386e+1 (1.97e+0) =

1.7895e+1 (4.64e+0) = 2.4627e+1 (5.78e+0) 1.8010e+1 (5.43e+0) 2.0877e+1 (6.57e+0) = 1.2844e+1 (3.98e+0) = 4.8735e-1 (3.31e-2) 6.2656e-1 (3.58e-2) 9.5868e-1 (4.99e-2) 1.0091e+0 (2.94e-2) 1.2535e+0 (5.39e-2) 1.9756e+2 (3.92e+1) = 2.9685e+2 (5.27e+1) 2.4472e+2 (4.56e+1) 3.1787e+2 (6.80e+1) 1.8970e+2 (3.07e+1) 6.6407e-1 (9.74e-2) = 9.6409e-1 (5.86e-2) 1.1388e+0 (7.23e-2) 1.1540e+0 (4.45e-2) = 1.2768e+0 (6.03e-2) 4.1716e-1 (5.48e-2) 4.4027e-1 (4.95e-2) 5.4680e-1 (8.02e-2) 3.6567e-1 (4.48e-2) = 5.3773e-1 (9.29e-2) 8.2496e+0 (3.02e-1) 8.2157e+0 (2.61e-1) 8.5033e+0 (3.14e-1) 8.3887e+0 (1.11e-1) = 8.7064e+0 (2.50e-1) 7.1734e+0 (7.08e-1) 1.3875e+1 (1.64e+0) 2.2819e+1 (2.09e+0) 2.8812e+1 (2.09e+0) =

2.0008e+1 (5.07e+0) = 2.2477e+1 (5.96e+0) = 2.2897e+1 (6.35e+0) 2.2935e+1 (7.63e+0) = 1.8375e+1 (6.88e+0) = 4.0968e-1 (3.59e-2) 5.7110e-1 (2.69e-2) 8.7703e-1 (2.93e-2) 9.9564e-1 (3.08e-2) = 1.1913e+0 (2.90e-2) 1.7634e+2 (2.28e+1) = 2.9640e+2 (5.60e+1) 2.7896e+2 (5.74e+1) 3.2520e+2 (5.38e+1) 2.5203e+2 (6.22e+1) 8.0256e-1 (1.51e-1) 9.5326e-1 (6.54e-2) 1.2622e+0 (9.95e-2) 1.1599e+0 (4.75e-2) = 1.3596e+0 (7.07e-2) 3.2036e-1 (4.27e-2) 3.2508e-1 (4.29e-2) 3.5195e-1 (3.68e-2) 3.8305e-1 (4.24e-2) = 3.5974e-1 (5.57e-2) 7.9412e+0 (2.65e-1) 7.9948e+0 (1.82e-1) 8.0924e+0 (2.20e-1) 8.4025e+0 (1.01e-1) = 8.0504e+0 (2.78e-1) 7.8028e+0 (7.25e-1) 1.3977e+1 (9.75e-1) 2.4337e+1 (1.69e+0) 2.9195e+1 (1.80e+0) =

1.5405e+1 (7.28e+0) = 2.5247e+1 (7.01e+0) 2.0949e+1 (4.51e+0) 2.2796e+1 (7.36e+0) 1.7091e+1 (7.59e+0) = 3.9867e-1 (3.98e-2) 6.5044e-1 (2.94e-2) 9.9447e-1 (5.37e-2) 9.9722e-1 (3.80e-2) 1.2309e+0 (5.97e-2) 6.9629e+1 (5.91e+1) + 2.1281e+2 (3.76e+1) 1.6173e+2 (5.28e+1) 2.2845e+2 (2.56e+1) = 1.5725e+2 (5.44e+1) = 6.8917e-1 (6.88e-2) = 8.4027e-1 (5.05e-2) 9.7037e-1 (4.21e-2) 1.1442e+0 (4.77e-2) = 1.1252e+0 (4.87e-2) 2.5622e-1 (4.81e-2) 3.2078e-1 (4.14e-2) 3.4703e-1 (5.54e-2) 3.6878e-1 (4.07e-2) = 3.8563e-1 (5.91e-2) 4.6484e+0 (7.97e-1) 4.6599e+0 (6.20e-1) 5.3118e+0 (8.42e-1) 8.3799e+0 (1.02e-1) = 5.7652e+0 (4.57e-1) 8.0618e+0 (9.20e-1) 1.2946e+1 (1.71e+0) 2.2599e+1 (2.24e+0) 2.9267e+1 (1.57e+0) =

2.0770e+1 (7.69e+0) = 2.7367e+1 (7.84e+0) 2.1660e+1 (6.37e+0) 2.9078e+1 (7.57e+0) 2.4259e+1 (6.67e+0) 4.1266e-1 (3.24e-2) 6.0372e-1 (3.08e-2) 8.9249e-1 (3.20e-2) 9.8697e-1 (3.45e-2) = 1.1972e+0 (4.72e-2) 1.9712e+2 (2.50e+1) = 3.3659e+2 (7.27e+1) 2.6831e+2 (4.96e+1) 3.3861e+2 (5.09e+1) 2.5810e+2 (6.28e+1) 8.1736e-1 (1.28e-1) 9.5835e-1 (8.51e-2) 1.0884e+0 (6.24e-2) 1.1410e+0 (4.40e-2) = 1.2298e+0 (4.69e-2) 3.4395e-1 (4.01e-2) 3.4887e-1 (4.31e-2) 3.7368e-1 (4.49e-2) 3.8903e-1 (3.84e-2) = 3.9898e-1 (4.53e-2) 7.9714e+0 (2.32e-1) 7.9936e+0 (2.96e-1) 8.1318e+0 (1.80e-1) 8.3268e+0 (7.90e-2) = 8.0532e+0 (1.93e-1) 8.1277e+0 (6.33e-1) 1.3917e+1 (1.33e+0) 2.3091e+1 (2.13e+0) 2.8480e+1 (2.91e+0) =

1.7893e+1 (7.07e+0) 2.0142e+1 (7.58e+0) 1.3286e+1 (4.79e+0) 1.9551e+1 (4.71e+0) 1.6146e+1 (6.50e+0) 2.8836e-1 (2.66e-2) 3.5430e-1 (2.29e-2) 6.7139e-1 (3.91e-2) 9.7883e-1 (3.39e-2) 1.0214e+0 (3.19e-2) 1.6229e+2 (5.40e+1) 1.6184e+2 (4.94e+1) 1.1701e+2 (5.20e+1) 2.1830e+2 (1.81e+1) 1.3391e+2 (5.19e+1) 7.0822e-1 (1.24e-1) 6.7576e-1 (7.19e-2) 9.3163e-1 (6.44e-2) 1.1429e+0 (3.29e-2) 1.0584e+0 (5.70e-2) 1.9895e-1 (2.49e-2) 1.5116e-1 (2.92e-2) 1.9403e-1 (4.04e-2) 3.7157e-1 (3.76e-2) 2.4778e-1 (5.29e-2) 2.7625e+0 (7.27e-1) 2.6572e+0 (5.34e-1) 3.3279e+0 (9.95e-1) 8.3752e+0 (1.32e-1) 3.3348e+0 (6.59e-1) 5.9104e+0 (1.15e+0) 9.3352e+0 (1.73e+0) 1.7097e+1 (2.66e+0) 2.8741e+1 (2.29e+0)

4.8556e+1 (4.06e+0) -

4.5093e+1 (3.80e+0) -

4.6884e+1 (2.53e+0) -

4.5270e+1 (4.36e+0) -

4.7442e+1 (3.46e+0) -

4.0533e+1 (4.74e+0)

1/35 0/31/4

3/35 0/26/9

0/35 0/25/10

2/35 1/25/9

2/35 0/28/7

27/35 —

DTLZ2

DTLZ3

DTLZ4

DTLZ5

DTLZ6

DTLZ7

15 Best/All Better/Worse/Similar

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Problem

267

268

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Table 9 Comparison results of the HMaPSO and NMPSO on the DTLZ for IGD values. Problem

M

NMPSO

HMaPSO

DTLZ1

3 5 8 10 15 3 5 8 10 15 3 5 8 10 15 3 5 8 10 15 3 5 8 10 15 3 5 8 10 15 3 5 8 10

2.7857e+1 (4.57e+0) 2.7223e+1 (4.22e+0) 2.2751e+1 (5.07e+0) 2.4756e+1 (4.24e+0) 2.4800e+1 (6.21e+0) 3.1765e-1 (3.90e-2) 4.8839e-1 (1.77e-2) 8.0178e-1 (3.30e-2) 1.0069e+0 (3.68e-2) = 1.1206e+0 (3.21e-2) 1.8890e+2 (2.90e+1) = 2.0946e+2 (8.54e+0) 2.0862e+2 (1.01e+1) 2.0719e+2 (1.91e+1) = 2.0932e+2 (2.32e+1) 7.1878e-1 (1.25e-1) = 8.1191e-1 (7.83e-2) 9.6900e-1 (5.11e-2) = 1.1572e+0 (4.45e-2) = 1.1244e+0 (5.08e-2) 2.2743e-1 (3.91e-2) = 2.3725e-1 (2.88e-2) 2.7173e-1 (4.71e-2) 3.7012e-1 (3.21e-2) = 2.8738e-1 (3.79e-2) = 1.9228e+0 (7.26e-1) + 2.8588e+0 (7.40e-1) = 3.3968e+0 (5.58e-1) = 8.3771e+0 (1.10e-1) = 3.1005e+0 (6.46e-1) + 4.9350e+0 (1.83e+0) = 9.2257e+0 (2.04e+0) = 1.7995e+1 (2.81e+0) 2.8790e+1 (2.45e+0) =

1.5288e+1 (7.29e+0) 1.7904e+1 (7.08e+0) 1.6483e+1 (5.35e+0) 1.6614e+1 (4.32e+0) 1.3669e+1 (5.87e+0) 2.7721e-1 (2.67e-2) 3.4426e-1 (2.17e-2) 6.9816e-1 (4.22e-2) 9.9208e-1 (3.97e-2) 1.0277e+0 (3.78e-2) 1.8606e+2 (3.85e+1) 1.7473e+2 (4.04e+1) 1.2350e+2 (4.98e+1) 2.1303e+2 (1.90e+1) 1.4131e+2 (4.43e+1) 7.3652e-1 (1.63e-1) 6.9865e-1 (5.62e-2) 9.1726e-1 (8.01e-2) 1.1610e+0 (4.66e-2) 1.0488e+0 (5.85e-2) 2.1057e-1 (3.49e-2) 1.4517e-1 (2.06e-2) 2.0290e-1 (4.18e-2) 3.7028e-1 (4.16e-2) 2.6375e-1 (5.37e-2) 2.5658e+0 (6.00e-1) 2.8771e+0 (7.45e-1) 3.5437e+0 (7.05e-1) 8.3483e+0 (1.02e-1) 3.8520e+0 (7.91e-1) 5.4672e+0 (1.11e+0) 8.7912e+0 (1.44e+0) 1.5305e+1 (3.50e+0) 2.8943e+1 (2.07e+0)

4.0272e+1 (7.29e+0) =

3.9923e+1 (5.42e+0)

10/35 2/17/16

25/35 —

DTLZ2

DTLZ3

DTLZ4

DTLZ5

DTLZ6

DTLZ7

15 Best/All Better/Worse/Similar

Fig. 3. Results of the six algorithms for the coal model.

Metric Indicator

NSGA-III

RVEA

HV SP Coverage

0.0000×10 (0.00×10 )= 3.3875×10-2 (5.32×10-3 ) = 1.0000×100 (0.00×100 ) 0

0

KnEA

0.0000×10 (0.00×10 )= 1.9759×10-2 (6.23×10-3 ) + 1.0000×100 (0.00×100 ) 0

0

MOEA/D-DE

0.0000×10 (0.00×10 )= 2.7564×10-2 (5.69×10-3 ) + 1.0000×100 (0.00×100 ) 0

0

PICEA-g

0.0000×10 (0.00×10 ) = 6.7726×10-2 (1.62×10-2 ) 9.6886×10-1 (2.80×10-2 ) 0

0

HMaPSO

0.0000×10 (0.00×10 ) = 2.7822×10-2 (1.01×10-2 ) + 1.0000×100 (0.00×100 ) 0

0

0.0000×100 (0.00×100 ) 3.6960×10-2 (1.39×10-2 ) 9.2627×10-1 (6.87×10-2 )

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Table 10 Result obtained using the six algorithms on the coal model with different indicators.

269

270

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this model. On the basis of the obtained nondominated solutions, the model can provide good choices for decision makers in regional planning. 5. Conclusions This paper proposes a hybrid many-objective PSO (HMaPSO) algorithm to solve the green coal production problem. Based on the process of coal production, a many-objective coal production model is established, in which five optimization objectives, including the coal economic, energy benefit, ecological, coal gangue economic, and social benefits are considered. To effectively solve this model, the proposed HMaPSO employs two strategies. First, the new offspring of the alternative pool is generated by employing different evolutionary operators. Second, the environmental selection mechanism is adopted to select and store the excellent solutions. To verify the effectiveness of HMaPSO, two series of experiments are performed in which the HMaPSO is verified on the DTLZ, and the HMaPSO is applied to solve the many-objective coal production model. The computational results show that the HMaPSO outperforms the NSGA-III, RVEA, KnEA, MOEA/D-DE, and PICEA-g on most test instances. The results also confirm the effectiveness of the proposed alternative pool and demonstrate the promising influence of the proposed approach. To further compare the multiple algorithms on this problem, the HV, coverage, and SP were considered as the performance metrics. In terms of the HV values, the six algorithms exhibited a similar performance. In terms of the coverage, the HMaPSO exhibited the best performance. For the SP values, the REVA exhibited the best performance, and the five remaining algorithms were ranked as follows: KnEA, PICEA-g, NSGA-III, HMaPSO, and MOEA/D-DE. In future work, we aim to improve the many-objective optimization model for green coal production problem and make it approach the actual production value. Moreover, other strategies can be used in the HMaPSO to achieve better performance. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Zhihua Cui: Writing - original draft. Jiangjiang Zhang: Conceptualization, Methodology. Di Wu: Data curation. Xingjuan Cai: Visualization, Investigation. Hui Wang: Writing - review & editing. Wensheng Zhang: Software, Validation. Jinjun Chen: Software, Validation. Acknowledgments This work is supported by National Key Research and Development Program of China (Grant no. 2018YFC16040 0 0), National Natural Science Foundation of China (Grant nos. 61806138, 61663028, U1636220, 61961160707, 61976212), Key R&D program of Shanxi Province (International Cooperation, Grant No. 201903D421048), Key R&D program of Shanxi Province (High Technology) under Grant No.201903D121119, the Distinguished Young Talents Plan of Jiangxi Province under Grant No. 20171BCB23075, Postgraduate education Innovation project of Shanxi province under of Grant No.2019SY495. References [1] Z.H. Cui, F. Xue, X.J. Cai, Y. Cao, G.G. Wang, J.J. Chen, Detection of malicious code variants based on deep learning, IEEE Trans. Ind. Inf. 14 (7) (2018) 3178–3196. [2] M. Ojha, K.P. Singh, P. Chakraborty, S. Verma, A review of multi-objective optimisation and decision making using evolutionary algorithms, Int. J. Bio-Inspir. Comput. 14 (2) (2019) 69–84. [3] E.E. Tsiropoulou, P. 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