In recent years, the optimization of multi-objective problems has focused on determining the distribution of the solution sets in the target space, rather than determining the Pareto front. Thus, appropriate solutions can be determined for multi-objective problems. This study uses the cluster mechanism, and expects that solutions in the repository can be dispersed in a non-overlapped Manner. Moreover, the differences between non-dominated solutions can be enhanced using the geometric features of the particle swarms in the target space. The cluster mechanism first selects a stored non-dominated solution as the first cluster center, and calculates the radius r of the circle. Other solutions within a distance r of the cluster center solution belong to the same cluster. The solution at the center of the circle is retained, and the other solutions in the cluster are removed under the assumption that they are very similar to the center solution. In this way, more diversified solutions are expected to be stored. As far as the image of information is concerned, the grey theory believes that the system presents the problematic or partial image elements. In respect of the scale of information, it points out that only an interval can be obtained, and that it is impossible to achieve accurate values in terms of the measurable scale of the system. These are also core problems to be addressed by the grey theory. In comparison with statistical methods, the grey theory is special in that the researcher cannot receive adequate information about the truth, but is still able to use the small quantity of information (small-sampled and abnormal) without a specific distribution in order to conduct the grey system theory analysis [8J.L. Deng, "Control problems of grey systems", Syst. Control Lett., vol. 1, pp. 288-294, 1982.
[http://dx.doi.org/10.1016/S0167-6911(82)80025-X] ].

Regarding the cluster radius r, we calculate the range of the non-dominated solution currently stored in the repository of n target functions, and obtain the minimal value of the n ranges. The minimal value is divided by n times the External Repository Size (ERS). The cluster radius r is expressed in Eq. (20). The cluster algorithm proceeds as follows:

• use the non-dominated solution currently stored in the repository to calculate the range of the target function, and use Eq. (20) to calculate the cluster radius r.
• the first solution in the repository is set as the center of the first cluster.
• the distance of the second solution from the current cluster center is computed. If it is less than the cluster radius r, the solution belongs to the same cluster; if it is greater than the cluster radius, the solution becomes the center of a new cluster.
• return to Step 3 until all solutions have been assigned to a cluster.
• for all clusters with more than one solution, remove redundant solutions using grey relational analysis to obtain the cluster center solution.

The degree of relationship among sub-systems or elements can be evaluated through grey relational analysis [8J.L. Deng, "Control problems of grey systems", Syst. Control Lett., vol. 1, pp. 288-294, 1982.
[http://dx.doi.org/10.1016/S0167-6911(82)80025-X] ], whereby the factors that influence the development trend are determined in order to learn the major features of the system. This process has the following steps. Grey relational analysis is supposed to calculate the measures among discrete series, but the data in the series must be consistent with the comparability of the series. Hence, it is necessary to pre-process the whole group of series in the form of statistical analysis. Such a process is called “grey relational generation”. The procedure is called normalization, and different factors can be taken into quantitative or qualitative (satisfaction) consideration [8J.L. Deng, "Control problems of grey systems", Syst. Control Lett., vol. 1, pp. 288-294, 1982.
[http://dx.doi.org/10.1016/S0167-6911(82)80025-X] ].

1. Normalize the original data by dividing the x_i (k) by the mean value of the sequence, as shown in Eq. (21).
2. Designate the standard sequence and calculate the difference sequence. Take the mean value as a standard sequence, i.e. , sequence 0; the difference sequence Δ_0i (k) indicates the absolute difference of elements k between sequence i and sequence 0, as expressed in Eq. (22).
3. Calculate the maximal difference Δ_max and minimal difference Δ_min, as expressed in Eqs. (23) and (24).
4. Calculate the grey relational coefficient γ_0i(k). The relational coefficient γ_0i(k) is defined in Eq. (25), x where is the adjustment factor.

The adjustment factor is ς, and ς [0,1]. In general, the value of the recognition coefficient is set as “0.5”, but it can be adjusted according to the actual needs to increase the difference in the result. When ς the is smaller, and the greater the correlation analysis is resolution [8J.L. Deng, "Control problems of grey systems", Syst. Control Lett., vol. 1, pp. 288-294, 1982.
[http://dx.doi.org/10.1016/S0167-6911(82)80025-X] ].

Step 5: Calculate the grey relationship Г_0i between each sequence and the standard sequence. The grey relationship Г_0i is defined in Eq. (26).

Step 6: Conduct sequencing according to the grey relationship.

Using a multi-objective problem with n targets as an example, Eq. (27) represents the coordinates of the circle center, where k represents the size of the external repository, w is the particle position, and F_n(.) is the nth target function.

After obtaining the circle center, we connect it to the non-dominated solutions in other repositories and the solutions determined by the particles. After the accumulation process described by Eq. (28), we obtain Eq. (29) to determine the angle between the two vectors.

where w_j is the solution determined by particle j, w_nk is the kth non-dominated solution in the external repository, and o_Fn is the assumed circle center according to Eq. (29).

After determining the angle between each particle and the non-dominated solutions in the repository, the particle selects the non-dominated solution with the smallest angle as the global optimal solution, and updates its velocity for the next iteration.

The CSP comprises a group of variables and constraints, where each variable has a given range. The constraint describes the correlation between variables, and CSP aims to determine the solution to all related constraints.

The CSP is an NP-complete problem, and thus appropriate solutions are generally lacking. Some scholars have proposed different methods to solve CSP. Constraint propagation has been used to reduce the size of the search space, backtracking has been directly applied to search for the probable solution, and a combined tree-search and consistent algorithm has been developed to effectively determine one or multiple applicable solutions. Nadel [47B. Nadel, "Tree search and arc consistency in constraint satisfaction algorithms", In: L. Kanal, and V. Kumar, Eds., Search in Artificial Intelligence., Springer-Verlag: New York, 1998, pp. 287-342.] compared the effectiveness of several such algorithms. The major difference is the consistency of the nodes in the course of solving the tree structure. The main characteristic of hybrid techniques is that, when a new value is assigned to a variable, all related constraints are examined, and all values that fail to meet the constraints are filtered so that the probable solution values remain consistent. If the probable range of these variables becomes empty, there is contradiction between the constraints, and the backtracking algorithm must be used. Constraint satisfaction problem (CSP) is one type of complex problems, and so it is difficult to find an appropriate solution to it. Scholars have put forward a wide range of solutions. Constraint propagation has been adopted to narrow the search space, and recalling has been applied to the search for possible solutions. Moreover, the literature has developed the combined tree-search and consistent algorithm to define one or more applicable solutions [9R. Kowalczyk, "Using constraint satisfaction in genetic algorithms", Proc. Australian New Zealand Conf. Intell. Inf. Syst., 1996pp. 272-275 , 10N. Barnier, and P. Brisset, "Optimization by hybridization of a genetic algorithm and constraint satisfaction techniques", In: Proc. IEEE World Congr. Comput. Intell. Evol. Comput., 1998, pp. 645-649.
[http://dx.doi.org/10.1109/ICEC.1998.700115] ].

Barnier and Brisset [10N. Barnier, and P. Brisset, "Optimization by hybridization of a genetic algorithm and constraint satisfaction techniques", In: Proc. IEEE World Congr. Comput. Intell. Evol. Comput., 1998, pp. 645-649.
[http://dx.doi.org/10.1109/ICEC.1998.700115] ] proposed a hybrid system that integrates a GA with a constraint satisfaction technique. This method solves the CSP to reduce the search space, and improves the computing efficiency of the GA by constraint satisfaction-based inference. Kowalczyk [9R. Kowalczyk, "Using constraint satisfaction in genetic algorithms", Proc. Australian New Zealand Conf. Intell. Inf. Syst., 1996pp. 272-275 ] proposed using the concept of constraint satisfaction in a GA. However, few studies have used constraint inference to improve the computational efficiency of PSO. Thus, there is no systematic description of how to integrate problem constraints and user requirements into a constraint network.

Some recent studies have combined the constraint inference mechanism with NSGAII. The experimental results showed that constraint inference was very effective at remedying the defects in NSGAII when solving complex problems. Therefore, we propose a hybrid method that combines PSO with NSGAII, and use particle filtering to accelerate the search. This overcomes the stagnation problem caused by complex problems or an increase in dimensionality. The particle filtering algorithm replaces the conditions meeting the constraints, but this produces an unsatisfactory rule. With the PSO computing technique and optimal solution search capability, the search space can be reduced, and the search can be accelerated. In traditional PSO updating, particles refer to the optimal solutions of individuals and groups, and weight is employed to update speed and then location. PSO and NSGAII use the adaptability function to evaluate the classification rules, and an increase in the problems’ complexity tends to lengthen the time of search, which can lead to partial optimization. Therefore, we proposed PSO and a hybrid and a new updating mechanism [10N. Barnier, and P. Brisset, "Optimization by hybridization of a genetic algorithm and constraint satisfaction techniques", In: Proc. IEEE World Congr. Comput. Intell. Evol. Comput., 1998, pp. 645-649.
[http://dx.doi.org/10.1109/ICEC.1998.700115] , 12C.A. Coello, G.T. Pulido, and M.S. Lechuga, "Handling multiple objectives with particle swarm optimization", IEEE Trans. Evol. Comput., vol. 8, pp. 256-279, 2004.
[http://dx.doi.org/10.1109/TEVC.2004.826067] , 18C.J. Lin, and M.H. Hsieh, "Classification of mental task from EEG data using neural networks based on particle swarm optimization", Neurocomputing, vol. 72, pp. 1121-1130, 2009.
[http://dx.doi.org/10.1016/j.neucom.2008.02.017] , 19C.K. Goh, K.C. Tan, D.S. Liu, and S.C. Chiam, "A competitive and cooperative co-evolutionary approach to multi-objective particle swarm optimization algorithm design", Eur. J. Oper. Res., vol. 202, pp. 42-54, 2010.
[http://dx.doi.org/10.1016/j.ejor.2009.05.005] ].

The architecture proposed in this paper is based on PSO and the particle filtering algorithm in the constraint-based PSO (CBPSO) module. To improve the effectiveness of the particle swarm, we propose two improved update mechanisms, one for PSO and one for NSGAII. Fig. (1) shows the research architecture. First, the PSO initiation module randomly generates particles according to the initial parameters. The particle filtering algorithm then filters the classification rule represented by each particle into reasonable and acceptable classification rules. The termination criterion is judged by calculating the fitness function. If the stop condition is not met, the velocity and position of each particle is updated, and the particle filtering algorithm checks the generated particles. In particle initiation and evolution, each particle must be filtered into a reasonable and acceptable form. When a particle is updated, its fitness is calculated, and P_id, P_gdare recorded. The set of local variables is then defined as fitness function, and the process is repeated until the constraints are met as shown in Fig. (1) [48M.J. Alves, and J.P. Costa, "An algorithm based on particle swarm optimization for multiobjective bilevel linear problems", Appl. Math. Comput., vol. 247, pp. 547-561, 2014.
[http://dx.doi.org/10.1016/j.amc.2014.09.013] -52N. Srinivas, and K. Deb, "Multi-objective optimization using non-dominated sorting in genetic algorithms", Evol. Comput., vol. 2, pp. 221-248, 1994.
[http://dx.doi.org/10.1162/evco.1994.2.3.221] ]. Some recent studies have integrated the constraint inference mechanism with NSGAII, and according to their results, the constraint inference is very effective in solving the problems in NSGAII. Therefore, we proposed a hybrid method that combined PSO with NSGAII and adopted the Particle Filter Algorithm to facilitate the search. This solved the stagnation caused by complex problems or the increase in dimension. The Particle Filter Algorithm substituted the conditions that met the constraints, but this led to unsatisfactory rules. PSO and the ability to search for the optimal solution can reduce the search space and facilitate the search, thus allowing the effective integration and application of Particle Filter Algorithm in the solution of some problems [10N. Barnier, and P. Brisset, "Optimization by hybridization of a genetic algorithm and constraint satisfaction techniques", In: Proc. IEEE World Congr. Comput. Intell. Evol. Comput., 1998, pp. 645-649.
[http://dx.doi.org/10.1109/ICEC.1998.700115] , 12C.A. Coello, G.T. Pulido, and M.S. Lechuga, "Handling multiple objectives with particle swarm optimization", IEEE Trans. Evol. Comput., vol. 8, pp. 256-279, 2004.
[http://dx.doi.org/10.1109/TEVC.2004.826067] , 18C.J. Lin, and M.H. Hsieh, "Classification of mental task from EEG data using neural networks based on particle swarm optimization", Neurocomputing, vol. 72, pp. 1121-1130, 2009.
[http://dx.doi.org/10.1016/j.neucom.2008.02.017] , 19C.K. Goh, K.C. Tan, D.S. Liu, and S.C. Chiam, "A competitive and cooperative co-evolutionary approach to multi-objective particle swarm optimization algorithm design", Eur. J. Oper. Res., vol. 202, pp. 42-54, 2010.
[http://dx.doi.org/10.1016/j.ejor.2009.05.005] ].

In the particles, the set of local variables U_p consists of two sets: one is the set with values (x₁, = v₁,...,x_p = v_p), and the other is the set without values (x_p+1, ..., x_m). Let U ₀ be the set of variables with values, and U_m be the set of all m variables with values.

In the architecture of CBPSO, each position value is limited during the particle construction stage. Thus, the initial swarm, as well as the swarms generated during the execution of the algorithm, is processed by constraint inference and the position selection mechanism. Basically, CBPSO uses particle filtering to execute constraint inference. The main technique is forward checking, whereby constraint information is used to determine acceptable position values, limit the effective range of the position values within the constraint network, and use related constraints to limit the range of the other positions.

In the particle filtering algorithm, the constraint set V must be satisfied within the constraint network (X, D, V). U_p is the set of local position variables with values, meaning the local position has a value.

Fig. (1) Hybrid PSO algorithm.

The particle filtering algorithm appoints a set containing a single constraint F, and this set is determined by U_p, from the set U ₀ without any position variables with values U_m to with complete position values. The process uses forward checking for ; then the following three results may be obtained:

1. No solution: meaning is inconsistent and Fis a contradictory allocation variable. F cannot be extended, which implies there is no position value for which an appropriate particle can be built, and an inappropriate particle is returned.
2. With solution: meaning one single solution has been found. The algorithm terminates, and all the allocated position variable values are returned (i.e., one acceptable particle is returned).
3. Unknown: meaning the particle filtering algorithm tries to create one single constraint to generate the position value of the next allocation and enlarge the set.

In terms of particle encoding, we assign one particle as a rule set. The data are divided into continuous and discrete sets. Discrete data contain two SubParticles: (1) a flag denoting whether the attribute is Enabled/Disabled; (2) an attribute value. Continuous data have three SubParticles: (a) a flag denoting whether the attribute is Enabled/Disabled; (b) an operator; and (c) a value. The operator is either “≥” or “<.” [46T.F. Liang, and H.W. Cheng, "Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method", J. Ind. Manag. Optim., vol. 7, pp. 365-383, 2011.
[http://dx.doi.org/10.3934/jimo.2011.7.365] -48M.J. Alves, and J.P. Costa, "An algorithm based on particle swarm optimization for multiobjective bilevel linear problems", Appl. Math. Comput., vol. 247, pp. 547-561, 2014.
[http://dx.doi.org/10.1016/j.amc.2014.09.013] ]

In the encoding mode, Enable/Disable represents whether the attribute occurs in the rule condition, where the range value lies between 0 and 1. The first generation is randomly generated. If the value is less than 0.5, the attribute does not occur (Disable) in the rule condition. If the value is from 0.5–1.0, the attribute occurs (Enable) in the rule condition. The operator is the encoding format of continuous attributes, and its range value is 0–1. If the range value is 0–0.4, the operator of this attribute is <; otherwise, it is ≥, and the generated attribute value will not exceed the range values of the various attributes [34S. Validi, A. Bhattacharya, and P.J. Byrne, "Integrated low-carbon distribution system for the demand side of a product distribution supply chain: A DoE-guided MOPSO optimiser-based solution approach", Int. J. Prod. Res., vol. 52, pp. 3074-3096, 2014.
[http://dx.doi.org/10.1080/00207543.2013.864054] , 35J.J. Jamian, M.N. Abdullah, H. Mokhlis, M.W. Mustafa, and A.H. Bakar, "Global particle swarm optimization for high dimension numerical functions analysis", J. Appl. Math., vol. 2014, pp. 1-14, 2014.
[http://dx.doi.org/10.1155/2014/329193] ].

In this paper, each particle represents a set of classification rules. Thus, the fitness function of PSO is evaluated by the rate of satisfied classification rules. If there are rules and attributes, the complete rule set is as below and shown in Fig. (2), [26M.A. Villalobos-Arias, G.T. Pulido, and C.A. Coello, "A new mechanism to maintain diversity in multi-objective metaheuristics", Optimization, vol. 61, pp. 823-854, 2012.
[http://dx.doi.org/10.1080/02331934.2010.534476] , 27F. Jolai, R. Tavakkoli-Moghaddam, and M. Taghipour, "A multi-objective particle swarm optimisation algorithm for unequal sized dynamic facility layout problem with pickup/drop-off locations", Int. J. Prod. Res., vol. 50, pp. 4279-4293, 2012.
[http://dx.doi.org/10.1080/00207543.2011.613863] , 30F.J. Cabrerizo, E. Herrera-Viedma, and W. Pedrycz, "A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts", Eur. J. Oper. Res., vol. 230, pp. 624-633, 2013.
[http://dx.doi.org/10.1016/j.ejor.2013.04.046] ]:

Fig. (2) Example for particle encoding.

IF con_1,1 And... And con_1,n Then Class=C₁

Else IF con_2,1 And... And con_2,n Then Class=C₂

Else IF con_i,1 And... And con_i,n Then Class=C_i

Else IF con_m,1 And... And con_m,n Then Class=C_m

Else Unknown Classification Results

Rule = {If A2 = 3 AND A3 < 3.86 THEN Classification =1}

where con_i,j is the jth attribute of the ith rule, and C_i is the classification result of the ith rule.

The algorithm terminates when some maximum number of iterations is reached or the improvement efficiency is less than 2% over a number of executions. The maximum number of iterations was set to 300 in this study [34S. Validi, A. Bhattacharya, and P.J. Byrne, "Integrated low-carbon distribution system for the demand side of a product distribution supply chain: A DoE-guided MOPSO optimiser-based solution approach", Int. J. Prod. Res., vol. 52, pp. 3074-3096, 2014.
[http://dx.doi.org/10.1080/00207543.2013.864054] , 35J.J. Jamian, M.N. Abdullah, H. Mokhlis, M.W. Mustafa, and A.H. Bakar, "Global particle swarm optimization for high dimension numerical functions analysis", J. Appl. Math., vol. 2014, pp. 1-14, 2014.
[http://dx.doi.org/10.1155/2014/329193] ].

In the traditional PSO update, the particle refers to individual and group optimum solutions, and uses a weight to update the velocity and then the position. As PSO and NSGAII use a fitness function to evaluate the classification rule, an increase in problem complexity tends to extend the search time, which may enable a local optimum to be attained. Thus, we propose novel update mechanisms for both PSO and the hybrid approach [46T.F. Liang, and H.W. Cheng, "Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method", J. Ind. Manag. Optim., vol. 7, pp. 365-383, 2011.
[http://dx.doi.org/10.3934/jimo.2011.7.365] -48M.J. Alves, and J.P. Costa, "An algorithm based on particle swarm optimization for multiobjective bilevel linear problems", Appl. Math. Comput., vol. 247, pp. 547-561, 2014.
[http://dx.doi.org/10.1016/j.amc.2014.09.013] ].

In the PSO equations, each rule attribute has its initial execution speed and position. When the updated velocity is obtained by Eq. (30), the position is updated according to Eq. (31). The rule set is then updated to reflect the updated position, the particle is compared with the optimal rule set, and the rule attribute value of the same position is retained while the particle position is updated again. If the position is better than the previous individual optimum, the value is updated to P_id; if it is better than the group optimum, its value is updated to P_gd [46T.F. Liang, and H.W. Cheng, "Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method", J. Ind. Manag. Optim., vol. 7, pp. 365-383, 2011.
[http://dx.doi.org/10.3934/jimo.2011.7.365] -48M.J. Alves, and J.P. Costa, "An algorithm based on particle swarm optimization for multiobjective bilevel linear problems", Appl. Math. Comput., vol. 247, pp. 547-561, 2014.
[http://dx.doi.org/10.1016/j.amc.2014.09.013] ].

The difference between our approach and traditional PSO is that the optimal rule set is compared with each particle, and the same rule set is retained. The purpose of this is to keep the features of the optimal rule set so that each particle can retain its advantages. Thus, we are more likely to obtain the optimal rule set. Next, the validity of the rule set is guaranteed by applying the particle filtering algorithm. The rule set is input into the particle filtering algorithm to determine acceptable classification rules, and the particle fitness is recalculated and updated [30F.J. Cabrerizo, E. Herrera-Viedma, and W. Pedrycz, "A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts", Eur. J. Oper. Res., vol. 230, pp. 624-633, 2013.
[http://dx.doi.org/10.1016/j.ejor.2013.04.046] , 34S. Validi, A. Bhattacharya, and P.J. Byrne, "Integrated low-carbon distribution system for the demand side of a product distribution supply chain: A DoE-guided MOPSO optimiser-based solution approach", Int. J. Prod. Res., vol. 52, pp. 3074-3096, 2014.
[http://dx.doi.org/10.1080/00207543.2013.864054] , 35J.J. Jamian, M.N. Abdullah, H. Mokhlis, M.W. Mustafa, and A.H. Bakar, "Global particle swarm optimization for high dimension numerical functions analysis", J. Appl. Math., vol. 2014, pp. 1-14, 2014.
[http://dx.doi.org/10.1155/2014/329193] ].

The hybrid update approach is illustrated in Fig. (3), where the particle for executing NSGAII is selected from the proportion function f generated at random in the old swarm. The randomly selected particle is updated by the NSGAII approach, and a crossover between P_Best and G_Best is applied according to the fitness values of the particles. The mutation operator converts Enable into Disable, and the operator converts ≥ to <. In contrast, it is randomly generated, and the better fitness value replaces the old particle as the new particle. A swarm containing higher fitness values is obtained by this update mode, and this process is repeated until the stop condition is met [30F.J. Cabrerizo, E. Herrera-Viedma, and W. Pedrycz, "A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts", Eur. J. Oper. Res., vol. 230, pp. 624-633, 2013.
[http://dx.doi.org/10.1016/j.ejor.2013.04.046] , 34S. Validi, A. Bhattacharya, and P.J. Byrne, "Integrated low-carbon distribution system for the demand side of a product distribution supply chain: A DoE-guided MOPSO optimiser-based solution approach", Int. J. Prod. Res., vol. 52, pp. 3074-3096, 2014.
[http://dx.doi.org/10.1080/00207543.2013.864054] , 35J.J. Jamian, M.N. Abdullah, H. Mokhlis, M.W. Mustafa, and A.H. Bakar, "Global particle swarm optimization for high dimension numerical functions analysis", J. Appl. Math., vol. 2014, pp. 1-14, 2014.
[http://dx.doi.org/10.1155/2014/329193] , 46T.F. Liang, and H.W. Cheng, "Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method", J. Ind. Manag. Optim., vol. 7, pp. 365-383, 2011.
[http://dx.doi.org/10.3934/jimo.2011.7.365] -48M.J. Alves, and J.P. Costa, "An algorithm based on particle swarm optimization for multiobjective bilevel linear problems", Appl. Math. Comput., vol. 247, pp. 547-561, 2014.
[http://dx.doi.org/10.1016/j.amc.2014.09.013] ].

Fig. (3) Hybrid approach.