An Improved Epsilon Constraint-handling Method in MOEA/D for CMOPs with Large Infeasible Regions

07/27/2017 ∙ by Zhun Fan, et al. ∙ 0

This paper proposes an improved epsilon constraint-handling mechanism, and combines it with a decomposition-based multi-objective evolutionary algorithm (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). The proposed constrained multi-objective evolutionary algorithm (CMOEA) is named MOEA/D-IEpsilon. It adjusts the epsilon level dynamically according to the ratio of feasible to total solutions (RFS) in the current population. In order to evaluate the performance of MOEA/D-IEpsilon, a new set of CMOPs with two and three objectives is designed, having large infeasible regions (relative to the feasible regions), and they are called LIR-CMOPs. Then the fourteen benchmarks, including LIR-CMOP1-14, are used to test MOEA/D-IEpsilon and four other decomposition-based CMOEAs, including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D. The experimental results indicate that MOEA/D-IEpsilon is significantly better than the other four CMOEAs on all of the test instances, which shows that MOEA/D-IEpsilon is more suitable for solving CMOPs with large infeasible regions. Furthermore, a real-world problem, namely the robot gripper optimization problem, is used to test the five CMOEAs. The experimental results demonstrate that MOEA/D-IEpsilon also outperforms the other four CMOEAs on this problem.

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1 Introduction

Real-world optimization problems usually involve the simultaneous optimization of multiple conflicting objectives with a number of constraints. Without loss of generality, a CMOP considered in this paper is defined as follows (Deb (2001)):

(1)

where is an

-dimensional objective vector,

is an inequality constraint, and is an equality constraint. is an -dimensional decision vector. The feasible region is defined as the set and .

In CMOPs, there are usually more than one constraint. The overall constraint violation is a widely used approach to deal with constraint violations, as it summarizes them into a single scalar, as follows:

(2)

If , is feasible; otherwise, it is infeasible. Any solution in set is feasible, and for any two solutions and , is said to dominate if for each and for at least one , denoted as . For a solution , if there is no other solution in dominating , then is called a Pareto optimal solution. A set including all of the Pareto optimal solutions is called a Pareto optimal set (). Mapping the into the objective space obtains a set of objective vectors, which is called a Pareto optimal front (), and .

CMOEAs aim to find a representative set of Pareto optimal solutions. They have to tackle the multiple conflicting objectives with a number of constraints simultaneously, and to maintain a good balance between convergence and diversity of the achieved solutions. In CMOEAs, there are two basic components: one is the constraint-handling mechanism, and the other is the multi-objective evolutionary algorithm (MOEA).

In terms of constraint-handling, many methods have been proposed in evolutionary optimization (Cai et al (2013); Hu et al (2013)). They can be roughly divided into penalty function methods, special representations and operators, repair methods, separation of objectives and constraints and hybrid methods (Coello (2002)). The penalty function method is widely used due to its simplicity in the constraint handling (Runarsson and Yao (2005)). However, the ideal penalty factors cannot be known in advance for an arbitrary CMOP, and tuning the penalty factors can be a very tedious task.

In recent years, a number of other constraint-handling techniques have had a relatively high impact in evolutionary optimization, including feasibility rules, stochastic ranking, -constrained method, novel penalty functions, novel special operators, multi-objective concepts and ensemble of constraint-handling techniques (Mezura-Montes and Coello Coello (2011)). However, most of them aim to solve constrained scalar optimization problems when they are first proposed.

MOEAs can be classified into three different types according to their selection approaches. The first type is non-dominated-based methods, and representative examples include NSGA-II (

Deb et al (2002)), PAES-II (Corne et al (2001)), SPEA-II (Zitzler et al (2001)), NSGA-III (Deb and Jain (2014)) and so on. The second type is decomposition-based approaches, and typical examples include MOEA/D (Zhang and Li (2007)), MOEA/D-DE (Li and Zhang (2009)), EAG-MOEA/D (Cai et al (2015)), MOEA/D-M2M (Liu et al (2014)), MOEA/D-SAS (Cai et al (2016)) and so on. Currently, MOEA/D is a popular algorithm to solve unconstrained multi-objective optimization problems (MOPs). MOEA/D decomposes a MOP into many scalar optimization subproblems, and optimizes them simultaneously in a collaborative way. The last type is indicator-based methods. This type of MOEAs selects solutions based on the improvement of a performance metric. Representative methods include IBEA (Zitzler and Künzli (2004)), SMS-EMOA (Beume et al (2007)), HypE (Bader and Zitzler (2011)), FV-MOEA (Jiang et al (2015)) and so on.

There are two commonly used test suites of CMOPs, including CTP (Deb (2001)) and CF test instances (Zhang et al (2008)

). For CTP1-CTP5 and CF1-CF10, the feasible regions are relatively large, and a CMOEA can approximate their PFs without encountering any infeasible obstacles during the entire evolutionary process. Thus, CTP1-5 and CF1-10 are not good test problems to evaluate the performance of constraint-handling mechanisms. For the remaining test problems CTP6-8, the feasible regions are relatively large, and the population of a CMOEA can reach these regions with high probability. Thus, CTP and CF test suites can not effectively measure the performance of constraint-handling techniques. When solving CTP (

Deb (2001)) and CF (Zhang et al (2008)) test instances, the constraint dominance principle (CDP) (Deb et al (2002)) is good enough to handle the constraints.

To overcome the shortcomings of the CTP and CF test suites discussed above, we propose a set of new CMOPs (named LIR-CMOP1-14). Each of them has a number of large infeasible regions, and the feasible regions are relatively small. The population of a CMOEA cannot easily discover these small feasible regions, which brings new challenges to the existing CMOEAs. In fact, many real-world optimization problems also have this characteristic. For example, the robot gripper optimization problem considered in this paper has large infeasible regions as illustrated in Section 6. Thus, it has important significance in practice to design specific mechanisms for solving CMOPs with large infeasible regions.

In this paper, we propose an improved -constrained version of MOEA/D to deal with CMOPs. Compared with the original -constrained method (Takahama and Sakai (2006)), the proposed method can keep a good balance in the search between the feasible and infeasible regions. It uses the information of the feasible ratio of the population to dynamically balance the exploration between the feasible regions and infeasible regions.

The remainder of the paper is organized as follows. Section 2 introduces related work on MOEA/D and the existing CMOEAs based on MOEA/D. Section 3 illustrates the improved epsilon constraint-handling method as here embedded in MOEA/D. Section 4 designs a set of new CMOPs (LIR-CMOPs) with large infeasible regions. Section 5 describes a comprehensive set of experiments to compare the proposed CMOEA (MOEA/D-IEpsilon) with four other CMOEAs, including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D. In Section 6, a robot gripper optimization problem is used to test MOEA/D-IEpsilon and the other four CMOEAs. Finally, Section 7 presents the conclusions.

2 Related work

2.1 Moea/d

MOEA/D (Zhang and Li (2007)) decomposes a MOP into a number of scalar optimization subproblems and optimizes them simultaneously in a collaborative way. Each subproblem is defined by a decomposition function with a weight vector . In MOEA/D, a set of uniformly spread weight vectors are adopted to formulate subproblems. The weight vectors satisfy and for each . In terms of decomposition methods, there are three commonly used approaches, including weighted sum (Miettinen (1999)), Tchebycheff (Miettinen (1999)) and boundary intersection approaches (Zhang and Li (2007)).

In the weighted sum approach, each subproblem is defined by summing each objective weighted by a different weight. The -th subproblem with the weighted sum decomposition method is defined as follows:

minimize (3)
subject to

For a minimizing MOP, in the case of a convex PF, the weighted sum approach can work well. However, if the PF is non-convex, only a part of PF can be found by this approach.

In the Tchebycheff decomposition method, the -th subproblem is defined as follows:

minimize (4)
subject to

where is the ideal point, and . The Tchebycheff method is a widely used decomposition approach. It can approximate both concave and convex parts of PFs.

In the boundary intersection approach, two distances and are defined to evaluate the convergence and diversity respectively. The -th subproblem is defined as follows:

minimize
subject to
where

The boundary intersection method is able to solve MOPs with any shape of PFs. However, the penalty factor must be set in advance.

2.2 Decomposition-based CMOEAs

In decomposition-based CMOEAs, a CMOP is decomposed into a set of constrained scalar optimization subproblems, and these subproblems are solved in a collaborative way simultaneously. Representative methods include C-MOEA/D (Asafuddoula et al (2012)), MOEA/D-Epsilon (Yang et al (2014)), MOEA/D-CDP (Jan and Khanum (2013)) and MOEA/D-SR (Jan and Khanum (2013)).

C-MOEA/D (Asafuddoula et al (2012)) embeds an epsilon constraint-handling approach into MOEA/D, and the epsilon value is set adaptively. To be more specific, the epsilon level is set to . denotes the mean value of the overall constraint violation in the current population, and () denotes the feasible ratio of solutions in the current population. For two solutions, if their overall constraint violations are both less than or their overall constraint violations are equal, the one with the better aggregation value is selected. Otherwise, the one with the smaller overall constraint violation is selected.

MOEA/D-Epsilon (Yang et al (2014)) also adopts the epsilon method to handle constraints. Unlike C-MOEA/D, the epsilon value in MOEA/D-Epsilon is set dynamically with the increase of generation counter . The detailed setting of the epsilon value can be found in (Takahama and Sakai (2006)).

MOEA/D-CDP (Jan and Khanum (2013)) adopts CDP (Deb et al (2002)) to deal with constraints in the framework of MOEA/D. There are three basic rules to select solutions. For two feasible solutions, the one with the better aggregation value is selected. For two infeasible solutions, the one with the smaller overall constraint violation is selected. For a feasible and an infeasible solution, the feasible one is selected.

MOEA/D-SR (Jan and Khanum (2013)) embeds the stochastic ranking method (SR) (Runarsson and Yao (2000)) in MOEA/D to deal with constraints. A parameter is set to balance the selection between the objectives and the constraints in MOEA/D-SR. For two solutions, if a random number is less than , the one with the better aggregation value is selected into the next generation. If the random number is greater than , the solutions selection is similar to that of MOEA/D-CDP. In the case of , MOEA/D-SR is equivalent to MOEA/D-CDP.

In summary, C-MOEA/D and MOEA/D-Epsilon both adopt the epsilon constraint-handling approach to solve CMOPs. To get across large infeasible regions, should be increased at sometimes, and be greater than the maximum overall constraint violation in the current population. However, in C-MOEA/D, is always less or equal than , and in MOEA/D-Epsilon, is always decreasing during the evolutionary process. In MOEA/D-CDP, feasible solutions are always better than infeasible solutions. Thus, the infeasible solutions which can help to get across large infeasible regions are difficult to survive. MOEA/D-SR applies a parameter to balance the searching between the feasible and infeasible regions. In order to get across large infeasible regions, should be set dynamically. However, is a static parameter in MOEA/D-SR. To overcome the shortcomings of the four decomposition-based CMOEAs discussed above, an improved epsilon constraint-handling method embedded in MOEA/D is proposed.

3 The Proposed method

In this section, the concept of epsilon level comparison, the original epsilon level setting method and the improved epsilon level setting approach are described.

3.1 Epsilon Level Comparison

In the epsilon constraint handling approach (Takahama and Sakai (2006)), the relaxation of constraints is controlled by the epsilon level . For two solutions and , their overall constraint violations are and . Then, for any satisfying , the epsilon level comparison is defined as follows:

(6)

In Eq. (6), the epsilon comparison approach is equivalent to CDP (Deb et al (2002)) when . In the case of , it does not consider any constraints. In other words, the comparison between any two solutions is based on their non-dominated ranks on objectives when .

3.2 Epsilon Level Setting

In the epsilon constraint-handling method, the setting of is quite critical. In (Takahama and Sakai (2006)), an epsilon level setting method is suggested as follows:

(7)

where is the top -th individual of the initial population sorted by overall constraint violations in a descending order. is to control the speed of reducing relaxation of constraints. is updated until the generation counter reaches the control generation . When , . The recommended parameter ranges in (Takahama and Sakai (2006)) are listed as follows: , and . denotes the population size, and represents the maximum evolutionary generation.

3.3 Improved Epsilon Level Setting

The setting of in Eq.(7) is always decreasing during the evolutionary process, which may not be suitable to solve CMOPs with large infeasible regions. To overcome this problem, an improved epsilon setting approach is suggested as follows:

(8)

where is the overall constraint violation of the top -th individual in the initial population, is the ratio of feasible solutions in the -th generation. ranges between and , and has two functions. One is to control the speed of reducing the relaxation of constraints, and the other is to control the scale factor multiplied by the maximum overall constraint violation. is to control the searching preference between the feasible and infeasible regions, and . is the maximum overall constraint violation found so far.

The setting method in Eq. 8 is sometimes the same as that in Eq. 7. If , in Eq. 7 is identically equal to zero, which tends to hinder a CMOEA’s exploration of the infeasible regions. However, in Eq. 8 is not identically equal to zero when according to the third rule of the proposed epsilon setting approach.

In the case , three rules are adopted to control the value of in Eq. 8. is adopted to strengthen the searching in the feasible regions. is used to strengthen the exploration in the infeasible regions. The last is same as in the CDP (Deb et al (2002)) constraint-handling method.

Two parameters and are applied to choose the right control rule for . If and , for setting is adopted. In this circumstance, is set to , which has an exponential decreasing rate. It has a faster descent rate than the epsilon setting in Eq. (7), which can help to enhance the searching in the feasible regions more effectively. If and , for setting is applied. In this situation, most solutions are feasible. Thus, strengthening the exploration in the infeasible regions may help a CMOEA to get across a number of large infeasible regions. In , , which strengthens the exploration in the infeasible regions. Thus, the improved epsilon method has the balanced ability to explore the feasible and infeasible regions simultaneously.

is a critical parameter to balance the searching between the feasible and infeasible regions. If the RFS is less than , is adopted to enhance the exploration in the feasible regions. Otherwise, is applied to enhance the exploration in the infeasible regions. Thus, the proposed epsilon constraint method can keep a good balance of exploration between the feasible and infeasible regions. It utilizes the RFS to dynamically balance the exploration between the feasible regions and infeasible regions.

Compared with the setting in Eq. (7), the proposed method in Eq. (8) has the ability to increase during the evolutionary process, which can help to solve CMOPs with large infeasible regions.

In the case of , is applied. In this situation, , and the epsilon constraint-handling method exerts the highest selection pressure toward the feasible regions.

3.4 Embedding the improved epsilon method in MOEA/D

The proposed MOEA/D-IEpsilon integrates the improved epsilon constraint-handling method in Eq. 8 into the framework of MOEA/D. In MOEA/D-IEpsilon, a CMOP is decomposed into a number of constrained scalar subproblems, and these subproblems are optimized simultaneously in a collaborative way. In our experimental studies, the Tchebycheff approach is adopted, and its detailed definition is listed in Eq. (4).

For a given weight vector , there exists an optimal solution of Eq. (4), and this optimal solution is also a Pareto optimal solution of Eq. (1). Therefore, we can achieve different Pareto optimal solutions of Eq. (1) by setting different weight vectors.

Input:
: the number of subproblems.
: the maximum generation.
weight vectors: .
: the size of the neighborhood.
: the selecting probability from neighbors.
: the maximal number of solutions replaced by a child.
Output: a set of feasible non-dominated solutions
1 Decompose a CMOP into subproblems associated with .
2 Generate an initial population .
3 Initialize according to Eq. (8).
4 Initialize the ideal point .
5 For each , set , where are the closest weight vectors to .
6 .
7 while  do
8       Set according to Eq. (8).
9       Generate a random permutation from .
10       for  to  do
11             Generate a random number .
12             .
13             if  then
14                  
15            else
16                  
17             end if
18            Generate through the DE operator.
19             Perform polynomial mutation on .
20             for  to  do
21                   if  then  ;
22                  
23             end for
24            Set .
25             while  or  do
26                   select an index from randomly.
27                   = UpdateSubproblems(, , )
28                   if  then  ;
29                  
30             end while
31            
32       end for
33      
34       Update according to Eq. (8)
35       = NondominatedSelect()
36 end while
Algorithm 1 MOEA/D-IEpsilon

The psuecode of MOEA/D-IEpsilon is listed in Algorithm 1. It is almost the same as that of MOEA/D, except for the method of subproblem updating. Lines 1-6 initialize a number of parameters in MOEA/D-IEpsilon. First, a CMOP is decomposed into subproblems which are associated with . Then the population , the initial epsilon value , the ideal point and the neighbor indexes are initialized.

Lines 11-22 generate a set of new solutions and update the ideal point . To be more specific, a set of solutions which may be updated by a newly generated solution is selected (lines 11-17). In line 18, the differential evolution (DE) crossover is adopted to generate a new solution . The polynomial mutation operator is executed to mutate in line 19. The ideal point is updated (lines 20-22).

Lines 23-30 implement the updating process of subproblems. In line 26, the subproblems are updated based on the improved epsilon constraint-handling approach, and the detailed procedures are listed in Algorithm 2. Finally, a set of non-dominated solutions () is selected based on the non-dominated sort in line 33.

1Function result = UpdateSubproblems(,,)
2       if  and  then
3             if  then
4                   =
5                  
6             end if
7            
8       else if  then
9             if  then
10                   =
11                  
12             end if
13            
14       else if  then
15             =
16            
17       end if
18      return
19 end
Algorithm 2 Subproblem Update

In Algorithm 2, there are three basic rules to update a subproblem. For two solutions and , if their overall constraint violations are less than or equal to , and has a smaller aggregation value (the value of the decomposition function) than that of , then is replaced by (lines 3-7). If and have the same overall constraint violation, and has a smaller aggregation value than that of , then is replaced by (lines 8-12). Otherwise, if has a smaller overall constraint violation than that of , then is replaced by (lines 13-14). When the subproblem is updated, the function returns , otherwise, it returns .

4 Test instances

To evaluate the performance of the proposed MOEA/D-IEpsilon, a set of new CMOPs with large infeasible regions (named LIR-CMOPs) is designed according to our previous work (Fan et al (2016)). In terms of constraint functions, all of them have large infeasible regions. In term of objective functions, there are two components: shape functions and distance functions (Huband et al (2006)).

The shape functions are applied to set the shape of the PFs. In the LIR-CMOP test suite, two types of shape functions, including both convex and concave shapes, are designed. Distance functions are adopted that test the convergence performance of a CMOEA. In LIR-CMOP5-14, the distance functions are multiplied by a scale factor, which increases difficulty of convergence. The detailed definitions of LIR-CMOPs are listed in the Appendix.

In this test suite, four test problems, including LIR-CMOP1-4, have large infeasible regions. Fig. 1(a)-(d) plot the feasible regions of LIR-CMOP1-4, respectively. It can be seen that the feasible regions of these test instances are very small. In other words, there are a number of large infeasible regions.

LIR-CMOP5 and LIR-CMOP6 have convex and concave PFs, respectively, as shown in Fig. 1(e)-(f) , and their PFs are the same as those of their unconstrained counterparts. The PFs of LIR-CMOP5 and LIR-CMOP6 can be achieved by a MOEA without any constraint-handling mechanisms.

In order to expand the test scope, LIR-CMOP7 and LIR-CMOP8 are designed. For these two test instance, their unconstrained PFs are located in the infeasible regions, and their PFs are situated on their constraint boundaries. Thus, a MOEA without constraint-handling methods cannot find the real PFs for LIR-CMOP7 and LIR-CMOP8, which are shown in Fig. 1(g)-(h).

LIR-CMOP9-12 have two different types of constraints. The first type creates large infeasible regions as shown in the black ellipses in Fig. 1(i)-(l). The second type creates difficulty in the entire objective space, as it divides the PFs of LIR-CMOP9-12 into a number of disconnected segments. For LIR-CMOP9-10, their PFs are a part of their unconstrained PFs, and for LIR-CMOP11-12, their PFs are situated on their constraint boundaries.

In the LIR-CMOP test suite, CMOPs with three objectives are also designed. Two CMOPs, including LIR-CMOP13 and LIR-CMOP14, have three objectives as shown in Fig. 2 (a)-(b) . The PF of LIR-CMOP13 is the same as that of its unconstrained counterpart. The PF of LIR-CMOP14 is located on the boundaries of its constraints.

(a) LIR-CMOP1 (b) LIR-CMOP2 (c) LIR-CMOP3
(d) LIR-CMOP4 (e) LIR-CMOP5 (f) LIR-CMOP6
(g) LIR-CMOP7 (h) LIR-CMOP8 (i) LIR-CMOP9
(j) LIR-CMOP10 (k) LIR-CMOP11 (l) LIR-CMOP12
Figure 1: Illustrations of the feasible and infeasible regions of LIR-CMOP1-12.
(a) LIR-CMOP13 (b) LIR-CMOP14
Figure 2: Illustrations of the infeasible regions of LIR-CMOP13-14.

5 Experimental study

5.1 Experimental Settings

To evaluate the performance of the proposed MOEA/D-IEpsilon, four other CMOEAs (MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D), with differential evolution (DE) crossover, are tested on LIR-CMOP1-14. The detailed parameters of these five CMOEAs are listed as follows:

  1. Mutation probability ( is the number of decision variables) and its distribution index is set to 20. , .

  2. Population size: . Neighborhood size: .

  3. Stopping condition: each algorithm runs for 30 times independently, and stops when 300,000 function evaluations are reached.

  4. Probability of selecting individuals in the neighborhood: .

  5. The maximal number of solutions replaced by a child: .

  6. Parameter setting in MOEA/D-IEpsilon: , , and .

  7. Parameter setting in MOEA/D-Epsilon: , , and .

  8. Parameter setting in MOEA/D-SR: .

5.2 Performance Metric

To measure the performance of MOEA/D-IEpsilon, C-MOEA/D, MOEA/D-CDP, MOEA/D-SR and MOEA/D-Epsilon, two commonly used metrics–the inverted generation distance (IGD) (Bosman and Thierens (2003)) and the hypervolume (Zitzler and Thiele (1999)) are adopted. The definition of IGD is shown next.

  • Inverted Generational Distance (IGD):

The IGD metric reflects the performance regarding convergence and diversity simultaneously. The detailed definition is as follows:

(9)

where is a set of representative solutions in the ideal PF, is an approximate PF achieved by a CMOEA. The value of IGD denotes the distance between and . For CMOPs with two objectives, 1000 points are sampled uniformly from the true PF to construct . (Note that this measure cannot be used if the true Pareto front is unknown, so it is used primarily for benchmarking purposes.) For CMOPs with three objectives, 10000 points are sampled uniformly from the PF to constitute . It is worth noting that a smaller value of IGD represents better performance with regards to both diversity and convergence.

  • Hypervolume ():

reflects the closeness of the non-dominated set achieved by a CMOEA to the real PF. The larger means that the corresponding non-dominated set is closer to the true PF.

(10)

where is the Lebesgue measure, is a reference point in the objective space. For a LIR-CMOP, the reference point is placed at 1.2 times the distance to the nadir point of the true PF. It is worth noting that a larger value of represents better performance regarding both diversity and convergence.

5.3 Discussion of Experiments

5.3.1 Performance comparison on LIR-CMOP test suite

The statistical results of the IGD values on LIR-CMOP1-14 achieved by five CMOEAs in 30 independent runs are listed in Table 1. As discussed in Section 4, LIR-CMOP1-4 have large infeasible regions in the entire search space. For these four test instances, MOEA/D-IEpsilon is significantly better than the other four tested CMOEAs in term of the IGD metric. Fig. 3(a)-(b) shows the final populations achieved by each CMOEA with the best IGD values during the 30 runs on LIR-CMOP1 and LIR-CMOP4. It is clear that MOEA/D-IEpsilon has the best performance regarding diversity among the five CMOEAs under test.

LIR-CMOP5-12 have large infeasible regions, as discussed in Section 4. It can be observed that MOEA/D-IEpsilon is significantly better than the other four tested CMOEAs on NCMOP5-12. The final populations achieved by each CMOEA on LIR-CMOP9 and LIR-CMOP11 with the best IGD values are plotted in Fig. 3(c)-(d). For LIR-CMOP9, MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D only achieve a part of the real PF. However, MOEA/D-IEpsilon can obtain the whole real PF. Thus, MOEA/D-IEpsilon performs better than the other four CMOEAs in terms of diversity. For LIR-CMOP11, the proposed method MOEA/D-IEpsilon can achieve the whole PF. However, the other four CMOEAs do not converge to the whole PF. Thus, MOEA/D-IEpsilon has better convergence performance than the other four CMOEAs. For three-objective test instances (LIR-CMOP13 and LIR-CMOP14), MOEA/D-IEpsilon is also significantly better than the other four CMOEAs.

Table 2 shows the results of the HV values of LIR-CMOP1-14 achieved by five CMOEAs in 30 independent runs. It is clear that MOEA/D-IEpilon is significantly better than the other four CMOEAs on all of the fourteen test instances in terms of the metric.

5.3.2 Analysis of Experimental Results

From the above performance comparison on the fourteen test instances LIR-CMOP1-14, it is clear that MOEA/D-IEpsilon has better diversity and convergence performance than the other four decomposition-based CMOEAs on these fourteen test instances. A common feature of these test instances is that each of them has a number of large infeasible regions, which demonstrates that the proposed epsilon constraint-handling method can deal with the large infeasible regions very well using its automatic adjustment of the epsilon level.

Test Instances MOEA/D-IEpsilon MOEA/D-Epsilon MOEA/D-SR MOEA/D-CDP C-MOEA/D
LIR-CMOP1 mean 7.213E-03 7.432E-02 1.719E-02 1.163E-01 1.290E-01
std 2.425E-03 3.538E-02 1.554E-02 7.265E-02 8.055E-02
LIR-CMOP2 mean 5.461E-03 6.407E-02 9.274E-03 1.244E-01 1.627E-01
std 1.520E-03 3.869E-02 9.723E-03 5.492E-02 5.819E-02
LIR-CMOP3 mean 1.117E-02 9.570E-02 1.792E-01 2.460E-01 2.751E-01
std 6.856E-03 4.529E-02 7.306E-02 4.444E-02 3.895E-02
LIR-CMOP4 mean 4.859E-03 6.141E-02 2.034E-01 2.486E-01 2.631E-01
std 1.591E-03 4.127E-02 6.038E-02 3.858E-02 3.331E-02
LIR-CMOP5 mean 2.107E-03 9.455E-01 1.041E+00 9.827E-01 8.637E-01
std 2.616E-04 4.705E-01 3.833E-01 4.140E-01 5.071E-01
LIR-CMOP6 mean 2.058E-01 1.177E+00 8.699E-01 1.224E+00 1.277E+00
std 4.587E-01 4.376E-01 5.992E-01 3.726E-01 2.587E-01
LIR-CMOP7 mean 4.598E-02 1.475E+00 1.074E+00 1.402E+00 1.511E+00
std 6.855E-02 5.309E-01 7.606E-01 6.226E-01 5.032E-01
LIR-CMOP8 mean 3.445E-02 1.522E+00 1.253E+00 1.361E+00 1.575E+00
std 6.002E-02 4.716E-01 6.597E-01 5.888E-01 3.849E-01
LIR-CMOP9 mean 1.290E-02 4.902E-01 4.883E-01 4.994E-01 4.902E-01
std 3.300E-02 4.221E-02 4.130E-02 2.526E-02 4.221E-02
LIR-CMOP10 mean 2.143E-03 2.202E-01 1.898E-01 2.042E-01 2.114E-01
std 1.261E-04 3.589E-02 6.277E-02 6.573E-02 5.641E-02
LIR-CMOP11 mean 4.713E-02 3.809E-01 2.911E-01 3.221E-01 3.321E-01
std 5.410E-02 1.131E-01 3.525E-02 7.723E-02 7.109E-02
LIR-CMOP12 mean 4.711E-02 2.574E-01 2.045E-01 2.289E-01 2.472E-01
std 5.662E-02 8.768E-02 6.771E-02 7.823E-02 8.883E-02
LIR-CMOP13 mean 6.447E-02 1.239E+00 1.059E+00 1.190E+00 1.215E+00
std 1.844E-03 2.555E-01 5.033E-01 3.290E-01 3.140E-01
LIR-CMOP14 mean 6.502E-02 1.172E+00 9.005E-01 1.204E+00 1.054E+00
std 1.635E-03 3.043E-01 5.455E-01 2.244E-01 4.515E-01

Wilcoxon’s rank sum test at a 0.05 significance level is performed between MOEA/D-IEpsilon and each of the other four CMOEAs. and denote that the performance of the corresponding algorithm is significantly worse than or better than that of MOEA/D-IEpsilon, respectively. The best mean is highlighted in boldface.

Table 1: IGD results of MOEA/D-IEpsilon and the other four CMOEAs on LIR-CMOP1-14 test instances
Test Instances MOEA/D-IEpsilon MOEA/D-Epsilon MOEA/D-SR MOEA/D-CDP C-MOEA/D
LIR-CMOP1 mean 1.015E+00 9.413E-01 9.840E-01 7.499E-01 7.344E-01
std 1.490E-03 3.751E-02 4.630E-02 1.202E-01 1.269E-01
LIR-CMOP2 mean 1.348E+00 1.267E+00 1.337E+00 1.093E+00 1.033E+00
std 1.717E-03 5.526E-02 2.252E-02 1.016E-01 9.522E-02
LIR-CMOP3 mean 8.686E-01 7.964E-01 5.892E-01 5.034E-01 4.715E-01
std 3.373E-03 3.618E-02 1.105E-01 5.141E-02 3.786E-02
LIR-CMOP4 mean 1.093E+00 1.024E+00 8.048E-01 7.397E-01 7.203E-01
std 1.910E-03 5.903E-02 8.956E-02 5.264E-02 4.480E-02
LIR-CMOP5 mean 1.461E+00 2.833E-01 1.773E-01 2.428E-01 3.870E-01
std 9.488E-04 5.766E-01 4.619E-01 5.031E-01 6.151E-01
LIR-CMOP6 mean 9.412E-01 1.255E-01 3.341E-01 8.582E-02 3.750E-02
std 3.848E-01 3.325E-01 4.458E-01 2.707E-01 1.446E-01
LIR-CMOP7 mean 2.847E+00 3.516E-01 9.943E-01 4.811E-01 2.933E-01
std 2.205E-01 9.304E-01 1.268E+00 1.083E+00 8.776E-01
LIR-CMOP8 mean 2.905E+00 2.690E-01 7.043E-01 5.223E-01 1.788E-01
std 2.103E-01 8.100E-01 1.094E+00 9.669E-01 6.669E-01
LIR-CMOP9 mean 3.692E+00 2.737E+00 2.733E+00 2.705E+00 2.737E+00
std 6.318E-02 1.484E-01 1.284E-01 8.883E-02 1.483E-01
LIR-CMOP10 mean 3.241E+00 2.874E+00 2.929E+00 2.899E+00 2.886E+00
std 3.537E-04 7.851E-02 1.064E-01 1.207E-01 1.126E-01
LIR-CMOP11 mean 4.263E+00 3.218E+00 3.479E+00 3.406E+00 3.386E+00
std 1.685E-01 3.542E-01 1.252E-01 2.135E-01 1.831E-01
LIR-CMOP12 mean 5.552E+00 4.858E+00 5.059E+00 4.972E+00 4.902E+00
std 1.730E-01 3.280E-01 2.103E-01 2.596E-01 3.233E-01
LIR-CMOP13 mean 5.710E+00 3.097E-01 1.184E+00 5.320E-01 4.642E-01
std 1.084E-02 1.048E+00 2.250E+00 1.442E+00 1.426E+00
LIR-CMOP14 mean 6.184E+00 5.617E-01 1.912E+00 4.032E-01 1.162E+00
std 1.053E-02 1.540E+00 2.705E+00 1.127E+00 2.281E+00

Wilcoxon’s rank sum test at a 0.05 significance level is performed between MOEA/D-IEpsilon and each of the other four CMOEAs. and denotes that the performance of the corresponding algorithm is significantly worse than or better than that of MOEA/D-IEpsilon, respectively. The best mean is highlighted in boldface.

Table 2: HV results of MOEA/D-IEpsilon and the other four CMOEAs on LIR-CMOP1-14 test instances
(a) LIR-CMOP1 (b) LIR-CMOP4 (c) LIR-CMOP9 (d) LIR-CMOP11
Figure 3: The non-dominated solutions achieved by each algorithm with the minimized IGD in the 30 independent runs for LIR-CMOP1, LIR-CMOP4, LIR-CMOP9 and LIR-CMOP11.

6 Robot Gripper Optimization

To verify the capability of MOEA/D-IEpsilon to solve real world optimization problems, a robot gripper optimization problem with two conflicting objectives and eight constraints is explored.

6.1 Definition of the robot gripper optimization

The robot gripper optimization problem is defined in (Saravanan et al (2009); Datta and Deb (2011)). Five objectives are formulated in these papers. The robot gripper optimization problem considered in this paper has two conflicting objectives and eight constraints. The geometrical structure of the gripper is plotted in Fig. 4.

Figure 4: The geometrical structure of robot gripper mechanism.

The robot gripper optimization problem considered in this paper is defined as follows:

(11)

where has seven decision variables, and each variable is shown in Fig. 4. The range of each decision variable is as follows: , , , , , and . Two rules are applied to fix the value of , and they are defined as follows:

According to the geometric analysis, the gripping force in Fig. 4 can be defined as follows:

(12)

The displacement of the gripper end is defined as follows:

(13)

where and denotes a dynamic displacement of the gripper actuator in the range of 0 to 100 mm.

The first objective represents a force transmission ratio between the actuating force and the minimum gripping force . We prefer to transform more actuating force into the gripper force. Thus, this objective should be minimized.

The second objective is the sum of all elements of the robot gripper. It is relevant to the weight of the robot gripper, and minimizing can lead to a lightweight design.

To study the distribution of solutions in the objective space for the robot gripper optimization problem, 3,000,000 solutions are generated, where 1,500,000 solutions are generated randomly, and the other 1,500,000 solutions are generated by MOEA/D-IEpsilon. In Fig. 5, we can observe that the robot gripper optimization problem has large infeasible regions (), which can be solved well by the proposed method MOEA/D-IEpsilon according to our previous analysis. To verify this hypothesis, MOEA/D-IEpislon and the other four decomposition-based CMOEAs are tested on the robot gripper optimization problems.

Figure 5: The distribution of solutions of the robot gripper optimization problem in the objective space, .
Symbol Meaning of parameter Value
Minimal dimension of 50mm
object to be gripped
Maximal range of the 150mm
gripper ends displacement
Maximal dimension of 100mm
object to be gripped
Maximal displacement of 100mm
the gripper actuator
Actuating force of the 100N
gripper
The lower bound of 50N
gripping force
Table 3: The parameters of the gripper optimization problem

6.2 Experimental study

6.2.1 Experimental settings

To solve the robot gripper optimization problem and evaluate the performance of the proposed MOEA/D-IEpsilon, five decomposition-based CMOEAs, including MOEA/D-IEpsilon, MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D with the differential evolution (DE) crossover, are tested on the robot gripper optimization problem. The detailed parameters of these five CMOEAs are the same as listed in Section 5.1 except for the number of function evaluations. In the case of the robot gripper optimization problem, each CMOEA stops when 600,000 function evaluations are reached. As the ideal PF of the gripper optimization problem is not known in advance, we use only the hypervolume metric (Zitzler and Thiele (1999)) to measure the performance of the five tested CMOEAs. In the robot gripper optimization case, the reference point .

6.2.2 Analysis of experiments

Table 4 shows the statistical results of values of MOEA/D-IEpsilon and the other four CMOEAs on the robot gripper optimization problem. It is clear that MOEA/D-IEpsilon is significantly better than the other four CMOEAs. To further demonstrate the superiority of the proposed method MOEA/D-IEpsilon, the non-dominated solutions achieved by each CMOEA during the 30 independent runs are plotted in Fig. 6(a)-(e). The box plot of values of the five CMOEAs is shown in Fig. 6(f). From Fig. 6, we see that MOEA/D-IEpsion has better performance than the other four CMOEAs.

Test Instances MOEA/D-IEpsilon MOEA/D-Epsilon MOEA/D-SR MOEA/D-CDP cMOEA/D
mean 1.897E+03 1.891E+03 1.889E+03 1.869E+03 1.865E+03
std 3.510E+00 7.151E+00 9.839E+00 8.124E+00 9.048E+00

Wilcoxon’s rank sum test at a 0.05 significance level is performed between MOEA/D-IEpsilon and each of the other four CMOEAs. and denote that the performance of the corresponding algorithm is significantly worse than or better than that of MOEA/D-IEpsilon, respectively. The best mean is highlighted in boldface.

Table 4: HV results of MOEA/D-IEpsilon and the other four CMOEAs on the gripper optimization problem
(a) MOEA/D-IEpsilon (b) MOEA/D-Epsilon (c) MOEA/D-SR
(d) MOEA/D-CDP (e) C-MOEA/D (f) The box plots of each CMOEA
Figure 6: The non-dominated solutions achieved by each algorithm during the 30 independent runs are plotted in (a)-(e). In (f), the box plots of each CMOEA are plotted.
Figure 7: The non-dominated solutions achieved by MOEA/D-IEpsilon.

In order to verify the correctness of the optimization results of the robot gripper optimization problem, three representative individuals (A, B and C) are selected from the non-dominated solutions achieved by MOEA/D-IEpsilon as shown in Fig. 7. The configurations of the robot gripper mechanism at each point are also plotted in Fig. 7.

To measure the minimum gripping force , a spring with a large stiffness coefficient is set vertically at the end of the robot gripper during the simulation process. The spring force is regarded as the gripping force when the robot gripper is balanced by the spring. The simulation tool is ADAMS 2013, and the stiffness coefficient of the spring is N/m.

Table 5 shows the simulation results of the minimum gripping force with three different configurations of the robot gripper. The relative errors between the theoretical gripping forces and the simulated gripping forces are less than . Thus, we can conclude that the optimization results of the robot gripper optimization problem achieved by MOEA/D-IEpsilon are correct.

Sampled point The theoretical gripping force (N) The simulated result (N) Relative error
A 50.0000 50.0002 0.0004%
B 142.3168 142.4582 0.0994%
C 92.5285 92.5877 0.0639%
Table 5: The simulated results of the minimum gripping force with three different robot gripper configurations.

7 Conclusion

This paper proposes an improved epsilon constraint-handling method embedded in the framework of MOEA/D. A new CMOEA named MOEA/D-IEpsilon has been proposed. The comprehensive experimental results indicate that MOEA/D-IEpsilon has the ability to cross the large infeasible regions. Compared with the other four decomposition-based CMOEAs including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D, MOEA/D-IEpsilon has following advantages:

  • The performance of MOEA/D-IEpsilon is not sensitive to the initial epsilon value.

  • MOEA/D-IEpsilon has the ability to explore the feasible and infeasible regions simultaneously during the evolutionary process.

  • MOEA/D-IEpsilon utilizes the feasible ratio of the current population to dynamically balance the exploration between the feasible regions and infeasible regions. It keeps a good balance of the searching between infeasible and feasible regions.

  • MOEA/D-IEpsilon is suitable for solving CMOPs with large infeasible regions.

In terms of CMOPs, a new set of CMOPs named LIR-CMOP1-14 was designed and presented in this paper. A common feature of these test instances is that they have large infeasible regions. The experimental results show that MOEA/D-IEpsion is significantly better than the other four CMOEAs on this test suite. Thus, we hypothesize that MOEA/D-IEpsilon is better than the other four CMOEAs in solving CMOPs with large infeasible regions, in general. To demonstrate the capacity of MOEA/D-IEpsilon to solve real engineering problems, a robot gripper optimization problem with two conflicting objectives and eight constraints was used as a test problem. The experimental results also demonstrated that MOEA/D-IEpsilon outperformed the other four CMOEAs.

Proposed further work includes studying new constraint-handling mechanisms to solve CMOPs with different types of difficulty. One possible way is to collect more information about the working population, and utilize such information to guide a CMOEA to select appropriate constraint-handling methods in different evolutionary stages.

Acknowledgements.
This work was supported in part by the National Natural Science Foundation of China (NSFC) under grant 61300159, 61473241 and 61332002, by the Natural Science Foundation of Jiangsu Province of China under grant BK20130808, by the Project of Internation as well as Hongkong,Macao&Taiwan Science and Technology Cooperation Innovation Platform in Universities in Guangdong Province under grant 2015KGJH2014, by China Postdoctoral Science Foundation under grant 2015M571751, by the Science and Technology Planning Project of Guangdong Province of China under grant 2013B011304002, by Educational Commission of Guangdong Province of China under grant 2015KGJHZ014, by the Fundamental Research Funds for the Central Universities of China under grant NZ2013306, and by the Guangdong High-Level University Project “Green Technologies” for Marine Industries.

8 Appendix

In this section, the detailed definitions of LIR-CMOP1-14 are listed in Table 6.

Problem Objectives Constraints
LIR-CMOP1
LIR-d2
LIR-CMOP3
LIR-CMOP4
LIR-CMOP5
LIR-CMOP6
LIR-CMOP7