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Abstract

An improved multi-objective particle swarm optimization with time-varying parameter and follower bee search is proposed in this article. In this algorithm, the weight of personal best solution decreases gradually as the iteration continues. This approach eliminates the effect caused by its poorer quality compared to global best solution so that the convergence ability of the algorithm is improved. Furthermore, the follower bee search in artificial bee colony algorithm is introduced to strengthen the randomness of the algorithm and discover more non-dominated solutions. A comparative simulation study is carried out using internal raw ore dispatching in an iron mining group that contains multiple stopes and concentrating mills. The results show that the proposed algorithm can significantly improve convergence and diversity.

Keywords

Multi-objective particle swarm optimization time-varying parameter follower bee search raw ore dispatching

Introduction

Multi-objective problems exist extensively in industrial production and daily life. Since there are conflicts between each objective, a unique optimal solution does not exist in a multi-objective problem.¹ Therefore, the goal when analyzing a multi-objective problem is to find a set of non-dominated solutions rather than a single solution. A non-dominated solution is also called a Pareto solution and a set of them form the Pareto front in the objective space. Moreover, the solutions are required to be as close as possible to the real front, distributed uniformly and broad. Particle swarm optimization (PSO)² is more and more widely applied in various optimization problems, including complex multi-objective problem.³ The PSO that is used to solve a multi-objective problem is called the multi-objective particle swarm optimization (MOPSO). Recently, some scholars have improved the MOPSO by introducing chaotic sequences⁴ and mutations.⁵ There are numerous MOPSO applications, such as actuator designs,⁶ train scheduling for urban rail transit,⁷ electrical distribution systems,⁸ and the design of efficient automatic train operation (ATO) systems’ speed profiles for metro lines.⁹

An iron mine group contains a number of stopes and concentrating mills with years of development. The raw ore can be transported from any stope to any concentrating mill inside. For various reasons, different stopes and concentrating mills have different technical and economic indexes. The raw ore dispatching inside the group directly affects the iron concentration indexes. Therefore, the allocation of raw ore can be regarded as a multi-objective problem in which the decision variables are the amounts transported from each stope to each concentrating mill.

According to the interaction and variation between global best solution (gbest) and personal best solution (pbest) in the PSO search iteration, an improved PSO with time-varying parameter and the follower bee search is proposed and named TF-PSO (PSO based on time-varying parameter and follower bee search). In this method, the effect of pbest is weakened in the iteration process (as its quality is below gbest) in order to improve the solution quality. Moreover, the follower bee search is executed for expanding the search space, especially in later iterations. Consequently, the TF-PSO is transformed into TF-MOPSO (MOPSO based on time-varying parameter and follower bee search) by introducing external memory and its maintenance method, redesigning the selection operators and search range of the follower bee and updating pbest in order to solve the multi-objective problem. Moreover, the simulation studies and results show that the Pareto front obtained by the proposed algorithm has a more uniform distribution than the traditional MOPSO and other improved MOPSO that were recently reported.

An improved PSO with time-varying parameter and follower bee search

Analysis of the traditional PSO

PSO is a kind of swarm intelligence optimization algorithm that imitates the foraging behavior of a flock of birds. In this method, each bird is viewed as a particle, and the particle is driven by gbest and pbest to search for the optimal solution. The velocity and position updating equations are shown as follows

\begin{matrix} v_{ij} (τ + 1) = ω v_{ij} (τ) + c_{1} r_{1} (gbes t_{j} - x_{ij} (τ)) \\ + c_{2} r_{2} (pbes t_{ij} - x_{ij} (τ)) \end{matrix}

(1)

x_{ij} (τ + 1) = x_{ij} (τ) + v_{ij} (τ + 1)

(2)

where $ω$ , $c_{1}$ , and $c_{2}$ stand for inertia weight, individual experience coefficient, and social experience coefficient, respectively. $x_{i}$ and $v_{i}$ are the position and velocity of the ith particle. $pbes t_{i}$ is the position of the ith pbest, and $gbest$ is the position of gbest, and gbest is also viewed as non-dominated solution in multi-objective problem. $j$ is the dimension value, $τ$ is the current iteration number, and $r_{1}$ and $r_{2}$ are two random numbers in [0, 1].

Tian¹⁰ proved that the position of a single particle will approach $(c_{1} r_{1} gbest + c_{2} r_{2} pbest) / (c_{1} r_{1} + c_{2} r_{2})$ . Therefore, it is obvious that the search operation of PSO is always affected by gbest and pbest over the entire iteration. According to the mechanism of PSO, pbest has a worse solution quality than gbest. In addition, each pbest will gradually gather around gbest, and thus pbest lacks randomness and exploitive capacity at later stages of the iteration. Consequently, we conclude that the effect of pbest should keep decreasing as the iteration progresses. Furthermore, in order to improve the explorative capacity of the algorithm, the follower bee search operator is also needed.

Time-varying parameter

After the analysis above, a modification that gradually reduces the coefficient of pbest is necessary. The method decreases $c_{2}$ as the iteration number $τ$ increases, making it a time-varying parameter. The velocity updating equation of the improved algorithm is shown as follows

\begin{matrix} v_{ij} (τ + 1) = ω v_{ij} (τ) + c_{1} r_{1} (gbes t_{ij} - x_{ij} (τ)) \\ + α^{τ} c_{2} r_{2} (pbes t_{j} - x_{ij} (τ)) \end{matrix}

(3)

where $α$ is a constant in the range of [0, 1].

Follower bee search

The follower bee search is the operator that improves the search effectiveness of follower bees around leader bees in an artificial bee colony (ABC) algorithm.^11,12 It can effectively modify the randomness of the algorithm. This operator consists of two parts. One part is the selection of a leader bee selection based on roulette. It makes the leader bee with a higher quality attract more follower bees. The other part is that each follower bee stochastically searches around the selected leader bee. The one who finds the best honey source becomes the new leader bee. In the TF-PSO, a particle is treated as the leader bee in an ABC, and the particle with best fitness value has the higher probability of being chosen.

An improved MOPSO with time-varying parameter and follower bee search

Since the final goal in solving a multi-objective problem is to acquire a set of uniformly distributed non-dominated solutions, the following operators are also necessary for the MOPSO.

Maintaining the external memory

The set used to store non-dominated solutions is called external memory and its capacity is usually the same as the population of the MOPSO. When the number of non-dominated solutions exceeds the external memory capacity, partial non-dominated solutions need to be removed. Likewise, in order to further enhance the uniformity of non-dominated solutions, the more crowded non-dominated solutions based on Euclidean distance in the objective space will be preferentially removed. The maintenance steps are listed as follows:

Step 1. Sort all non-dominated solutions according to the fitness values of the first objective;

Step 2. Calculate the Euclidean distances between sorted non-dominated solutions based on the fitness values of all the objectives;

Step 3. Find the shortest distance, denoted as $d_{i}$ ;

Step 4. Compare $d_{i - 1}$ and $d_{i + 1}$ . If $d_{i - 1}$ shorter than $d_{i + 1}$ remove the ith non-dominated solution. Otherwise, remove the (i+1)th solution;

Step 5. Repeat Steps 2 through 4 until the number of non-dominated solutions is equal to the capacity of the external memory.

Calculating the selection probability and search range of the follower bee

In TF-MOPSO, non-dominated solutions are the followed objects of follower bees. Furthermore, the selection probability and search range of the follower bee calculation method are based on the average distance on both sides of the sorted non-dominated solution in order to make the non-dominated solutions spread as uniformly as possible. The chosen probability of each non-dominated solution is calculated by equation (4)

P_{i} = \frac{d r_{i}}{\sum d r_{i}}, i = 2, 3, \dots, gnum - 1

(4)

where $P_{i}$ is the chosen probability of ith non-dominated solution, $d r_{i} = (d_{i - 1} + d_{i}) / 2$ , $gnum$ is the capacity of the external memory, and $d_{i}$ is the Euclidean distance between ith non-dominated solution and (i+1)th non-dominated solution. Therefore, the more congested non-dominated solution has a lower chosen probability. Moreover, in order to make the more congested non-dominated solutions out of the crowded area, the follower bee search range is also calculated based on $d r_{i}$ . The calculation method is shown as follows

ra n_{i} = (\begin{matrix} \bar{d} & d r_{i} \geq \bar{d} \\ 2 \bar{d} - d r_{i} & d r_{i} < \bar{d} \end{matrix}, i = 2, 3, \dots, gnum - 1

(5)

where $\bar{d}$ is the mean of all distances. The search range for each dimension $ran g_{ij}$ is calculated by the following equation

\begin{array}{l} r a n g_{i j} = \frac{r a n_{i} (x_{j}^{m a x} - x_{j}^{m i n})}{\sum_{k = 1}^{g n u m - 1} d_{k}}, \\ i = 2, 3, …, g n u m - 1, j = 1, 2, …, D \end{array}

(6)

where $x_{j}^{\max}$ , $x_{j}^{\min}$ stand for upper and lower bounds, respectively, of the value of the $j$ dimension.

Updating the pbest of each particle

Each objective conflicts with each other in a multi-objective problem. Therefore, the fitness values of all the objective functions should be treated as the factors applied to compare the two solutions. Furthermore, because the value range of each objective is usually not the same as each other, even with several orders of magnitude, a criterion up is proposed in the article and used to compare two non-dominated solutions for updating the pbest of each particle

up = \sum_{i = 1}^{o} \frac{fitnes s_{i}}{maxfitnes s_{i}}

where $o$ stands for the number of objective and $fitnes s_{i}$ presents the current fitness value of ith objective. $maxfitnes s_{i}$ is the upper bound of the value of ith objective; it is unknown in the practical problem, and thus, it is replaced by the maximum value found by the single-objective PSO in this method.

The steps of TF-MOPSO

The complete TF-MOPSO consists of the operators mentioned above in a certain order, and the concrete procedure for implementing the proposed algorithm is given by the following steps:

Step 1. Initialize the positions and velocities of all the particles in the swarm;

Step 2. Calculate the fitness value of every objective for each particle, pick up the non-dominated solutions and store them to the external memory;

Step 3. Select a non-dominated solution from the external memory instead of gbest in equation (2) and update the positions and velocities of all the particles according to equations (2) and (3); update the pbest of each particle;

Step 4. Merge the new solutions obtained by the step above with the solutions in external memory and pick up the non-dominated solutions. If the number of non-dominated solutions exceeds the capacity of the external memory, execute the maintenance;

Step 5. Execute the follower bee search operator according to equations (4)–(6); update the pbest of each particle again;

Step 6. Execute Step 4 again;

Step 7. Return to Step 3 until the iteration runs up to the maximum number.

Raw ore dispatching problem in an iron mining group

The production of iron concentrate includes the two major steps of mining and concentration. The site of the mining operation is called the stope, and the site of concentration is called the concentrating mill. A mine group consists of five stopes and ore concentrating mills, and raw ore produced at the different mining sites can be sent to any concentrating mill for processing. Because of the differences in raw ore properties and mining technology, the raw ore indexes and prices of different stopes are also different. Similarly, due to the management level, process differences and other factors, production capacity and the cost of each concentrating mill, are different. Consequently, by adjusting the amount of raw ore transported from different stopes to different concentrating mills, the technical and economic indexes of iron concentrate production of the whole mine group can be changed. Combined with the constraints existing in the real production of the mine, a multi-objective optimization problem is formed.

The multi-objective optimization problem includes the two objectives of maximum output of concentrate in unit time and minimum variable cost of unit concentrate. Moreover, there are four constraints. The first constraint is the mill feed grade of each concentrating mill. Due to the requirements of the concentration process, the concentrating mill can only handle raw ore of a certain grade. The second constraint is the concentration ratio of the whole mining group. To ensure the effective utilization of raw ore, it must be less than a certain value. The third constraint is the mill feed grade of each concentrating mill. This means that the raw ore transported from each stope during the unit time cannot exceed its mining capacity and must be less than the requirements of the group. The last constraint is the processing capacity of each concentrating mill, which is similar to the constraint of the stopes.

Objective 1: maximum output of concentrate

Q (x) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \frac{x_{ij}}{D_{j} (x)} ton

where $x_{ij} (i = 1, 2, \dots, m; j = 1, 2, \dots, n)$ are decision variables that indicate the mount of raw ore that was transported from the ith stope to the jth concentrating mill. $i$ is the stope number and $j$ is the concentrating mill number. $D_{j} (x)$ is the concentration ratio of the jth concentrating mill according to the historical data. The regression equation is as follows

D {(x)}_{j} = \sqrt{a_{j} - \frac{p - w_{j}}{β_{j} (x) - w_{j}}}

where $a_{j}$ is constant, $p$ is concentrate grade, and $w_{j}$ is tailings grade of the jth concentrating mill

β_{j} (x) = \frac{\sum_{i = 1}^{m} β_{i} x_{ij}}{\sum_{i = 1}^{m} x_{ij}}

is the mill feed grade of the jth concentrating mill and $β_{i}$ is the head grade of the ith stope.

Objective 2: minimum variable cost of unit concentrate

C (x) = \frac{MC (x)}{Q (x)} yuan

where $MC (x)$ is the total variable cost and computed with the equation below

MC (x) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} r_{i} x_{ij} + \sum_{j = 1}^{n} \sum_{i = 1}^{m} e {(x)}_{j} x_{ij} + \sum_{i = 1}^{m} \sum_{j = 1}^{n} t_{ij} x_{ij}

and $r_{i}$ is the production cost of the ith raw ore, $t_{ij}$ is unit transportation cost of the raw ore from the ith stope to the jth concentrating mill, and $e_{j} (x)$ is the unit concentration cost of the jth concentrating mill, and its regression equation is shown as follows

e {(x)}_{j} = b_{j} \frac{p - w_{j}}{β_{j} (x) - w_{j}}

where $b_{j}$ is constant.

Constraint 1: mill feed grade of each concentrating mill

β_{L} \leq β_{j} (x) \leq β_{H}

where $β_{L}$ and $β_{H}$ are the lower and upper bounds, respectively, of the mill feed grade.

Constraint 2: concentration ratio of the whole mining group

D (x) = \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{n} x_{ij}}{Q (x)} \leq D_{H}

where $D_{H}$ is the upper bounds of concentration ratio.

Constraint 3: raw ore output of each stope

O L_{i} \leq \sum_{i = 1}^{m} \leq O H_{i}

where $O L_{i}$ and $O H_{i}$ are the lower and upper bounds, respectively, of the raw ore output of ith stope.

Constraint 4: processing capacity of each concentrating mill

P L_{j} \leq \sum_{j = 1}^{n} \leq P H_{j}

where $P L_{i}$ and $P H_{i}$ are the lower and upper bounds, respectively, of processing capacity of each concentrating mill;

The parameters required for the above model are listed in Tables 1 –4.

Table 1.

Comprehensive production parameters.

Index	Symbol	Value	Unit
Max mill feed grade	$β_{L}$	30	%
Min mill feed grade	$β_{H}$	40	%
Concentrate grade	$p$	65	%
Max concentration ratio	$D_{H}$	3.5	1

Table 2.

Raw ore parameters.

Stope	A	B	C	D	E
Head grade $(β_{i})$	36.5	36.5	36.0	35.5	37.0
Min output $(O L_{i})$	700	1100	1000	500	800
Max output $(O H_{i})$	800	1200	1100	600	900
Unit price $(r_{i})$	40.6	61.3	28.3	30.5	41.9

Table 3.

Concentrating mill parameters.

Concentrating mill	1	2	3	4	5
Parameter $(a_{j})$	8.4	10.0	11.7	13.6	8.7
Parameter $(b_{j})$	35.2	42.8	38.2	41.4	57.9
Min capacity $(P L_{j})$	700	1100	1000	500	800
Max capacity $(P H_{j})$	800	1200	1100	600	900
Tailings grade $(w_{j})$	16.6	15.7	18.6	19.2	16.9

Table 4.

Transportation cost.

Unit transportation cost $(t_{ij})$	A	B	C	D	E
Concentrating mill 1	0.11	0.13	1.97	5.66	7.96
Concentrating mill 2	2.41	1.09	0.09	3.79	6.24
Concentrating mill 3	3.25	1.75	0.08	3.40	5.93
Concentrating mill 4	5.78	4.25	2.83	0.09	2.96
Concentrating mill 5	8.34	6.97	5.93	2.63	0.09

The mining group studied in this article includes 5 stopes and 5 concentrated mills, which results in 25 raw ore transportation paths. The volume of the raw ore transported on these 25 paths are the decision variables of the dispatching problem. By integrating the objectives and constraints mentioned above, a 25-dimension bi-objective problem with numerous constraints is formed as the object of the TF-MOPSO and contrast algorithms.

Simulation

In this article, the traditional MOPSO, Multi-objective adaptive chaotic particle swarm optimization (MACPSO),⁴ and Multi-objective particle swarm optimization with two normal mutations (MN-PSO)⁵ are selected as the compared algorithms. The proposed method and the comparison algorithms are applied for internal raw ore dispatching in an iron mining group. All the simulations are performed on a personal computer (PC) with the Windows 7 operating system, VC++, an Intel® Core™ i3-330M Processor, and 3 GB of memory. The general parameters of the proposed algorithm are selected as ω = 0.7, a = 0.9, c1 = c2 = 1.35; the external memory and population number are 100, and the maximum number of iterations is 500.

The fitness values of the four pairs of endpoints are shown in Table 5, where $f_{1}$ and $f_{2}$ stand for the fitness values of the first objective (output of concentrate) and the second objective (variable cost of unit concentrate), respectively. The data indicate that the widths of the Pareto fronts found by the four algorithms are close. Moreover, the Pareto fronts obtained by the four algorithms are illustrated in Figure 1. It can be seen that the three Pareto fronts obtained by compared methods have nonuniform distribution obviously, dense in the middle and sparse in the both ends. From the corresponding figures, it can be concluded that the distribution of the Pareto front obtained by TF-MOPSO is the most uniform of the Pareto fronts. So, we can say that TF-MOPSO has the best performance in solving this internal raw ore dispatching in an iron mining group problem.

Table 5.

Simulation result.

Algorithm	Min $(f_{1})$	Min $(f_{2})$	Max $(f_{1})$	Max $(f_{2})$
MOPSO	1560.39	413.49	1648.72	415.81
MACPSO	1593.44	413.46	1650.06	415.67
MN-PSO	1599.53	413.45	1650.66	415.90
TF-MOPSO	1597.27	413.38	1650.06	415.69

MOPSO: multi-objective particle swarm optimization; TF-MOPSO: MOPSO based on time-varying parameter and follower bee search.

Figure 1.

Simulation results.

Conclusion

The main contributions of this article are a new improved MOPSO with time-varying parameter and follower bee search and its application in internal raw ore dispatching in an iron mining group. Contrasted simulation results with several algorithms show that the proposed algorithm has the best performance in convergence and distribution to the multi-objective problem with multiple constraints presented in this article. Future studies will focus on the theoretical research of the stability of the proposed algorithm in order to ensure its reliability.

Footnotes

Handling Editor: Gang Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Beijing Key Discipline Development Program (No. XK100080537).

References

Liu

Tian

. R2-MOPSO: a multi-objective particle swarm optimizer based on R2-indicator and decomposition. In: Proceedings of 2015 IEEE congress on evolutionary computation, Sendai, Japan, 25–28 May 2015, pp.3148–3155. New York: IEEE.

Clerc

Kennedy

. The particle swarm—explosion, stability, and convergence in a multidimensional complex space. IEEE T Evolut Comput 2002; 6: 58–73.

Curtis

Lewis

. The effect of bias on particle behaviour for MOPSO. In: Proceedings of the 2016 congress on evolutionary computation, Vancouver, BC, Canada, 24–29 July 2016, pp.711–718. New York: IEEE.

Yang

Che

et al . Multi-objective adaptive chaotic particle swarm optimization algorithm. Cont Decis 2015; 30: 2168–2174.

Gao

et al . Multi-objective particle swarm optimization with two normal mutations. Cont Decis 2015; 30: 939–942.

Liang

Zhang

You

et al . Multi-objective Gaussian particle swarm algorithm optimization based on niche sorting for actuator design. Adv Mech Eng 2015; 7: 1–7.

Chu

Zhanga

Chen

et al . Pareto optimal train scheduling for urban rail transit using generalized particle swarm optimization. Adv Mech Eng 2016; 8: 1–15.

Sahoo

Ganguly

Das

. Simple heuristics-based selection of guides for multi-objective PSO with an application to electrical distribution system planning. Eng Appl Artif Intel 2011; 24: 567–585.

Domnguez

Fernndez-Cardador

Cucala

. Multi objective particle swarm optimization algorithm for the design of efficient ATO speed profiles in metro lines. Eng Appl Artif Intel 2014; 29: 567–585.

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