Difficulty Adjustable and Scalable Constrained Multi-objective Test Problem Toolkit

12/21/2016 ∙ by Zhun Fan, et al. ∙ 0

Multi-objective evolutionary algorithms (MOEAs) have achieved great progress in recent decades, but most of them are designed to solve unconstrained multi-objective optimization problems. In fact, many real-world multi-objective problems usually contain a number of constraints. To promote the research of constrained multi-objective optimization, we first propose three primary types of difficulty, which reflect the challenges in the real-world optimization problems, to characterize the constraint functions in CMOPs, including feasibility-hardness, convergence-hardness and diversity-hardness. We then develop a general toolkit to construct difficulty adjustable and scalable constrained multi-objective optimization problems (CMOPs) with three types of parameterized constraint functions according to the proposed three primary types of difficulty. In fact, combination of the three primary constraint functions with different parameters can lead to construct a large variety of CMOPs, whose difficulty can be uniquely defined by a triplet with each of its parameter specifying the level of each primary difficulty type respectively. Furthermore, the number of objectives in this toolkit are able to scale to more than two. Based on this toolkit, we suggest nine difficulty adjustable and scalable CMOPs named DAS-CMOP1-9. To evaluate the proposed test problems, two popular CMOEAs - MOEA/D-CDP and NSGA-II-CDP are adopted to test their performances on DAS-CMOP1-9 with different difficulty triplets. The experiment results demonstrate that none of them can solve these problems efficiently, which stimulate us to develop new constrained MOEAs to solve the suggested DAS-CMOPs.

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I Introduction

Practical optimization problems usually involve simultaneous optimization of multiple and conflicting objectives with many constraints. Without loss of generality, constrained multi-objective optimization problems (CMOPs) can be defined as follows:

minimize (1)
subject to

where is a

-dimensional objective vector,

defines -th of inequality constraints, defines -th of equality constraints. If is greater than three, we usually call it a constrained many-objective optimization problem (CMaOP).

A solution is said to be feasible if it meets and at the same time. For two feasible solutions and , solution is said to dominate if for each and for at least one , denoted as . For a feasible solution , if there is no other feasible solution dominating , is said to be a feasible Pareto-optimal solution. The set of all the feasible Pareto-optimal solutions is called Pareto Set (). Mapping the into the objective space results in a set of objective vectors, denoted as the Pareto Front (), where .

For CMOPs, more than one objective need to be optimized simultaneously subject to constraints. Generally speaking, CMOPs are much more difficult to solve than their unconstrained counterparts - unconstrained multi-objective optimization problems (MOPs). Constrained multi-objective evolutionary algorithms (CMOEAs) are particularly designed to solve CMOPs, with the capability of balancing the search between the feasible and infeasible regions in the search space [1]. In fact, two basic issues need to be considered carefully when designing a CMOEA. One is to balance the feasible solutions and the infeasible solutions, the other is to balance the convergence and diversity of a CMOEA.

To address the former issue, constraint handling mechanisms need to be carefully designed by researchers. The existing constraint handling methods can be broadly classified into five different types, including feasibility maintenance, use of penalty functions, separation of constraint violation and objective values, multi-objective constraint handling and hybrid methods

[2].

The feasibility maintenance methods usually adopt special encoding and decoding techniques to guarantee that a newly generated solution is feasible. The penalty function-based method is one of the most popular approaches. The overall constraints violation is added to each objective with a predefined penalty factor, which indicates a preference between the constraints and the objectives. The penalty function-based method includes static penalties [3], dynamic penalties [4], death penalty functions [3], co-evolutionary penalty functions [5], adaptive penalty functions [6, 7, 8] and self-adaptive penalty functions [9, 10] etc. In the methods using separation of constraint violation and objective values, the constraint functions and the objective functions are treated separately. Variants of this type include stochastic ranking (SR) [11], constraint dominance principle (CDP) [12], epsilon-constrained methods [13, 14]. In the multi-objective constraint handling method, the constraint functions are transformed to one extra objective function. Representative methods of this type include infeasibility driven evolutionary algorithm (IDEA) [15], COMOGA[16] and Cai and Wang’s Method (CW) [17], etc. The hybrid methods of constraint handling usually adopt several constraint-handling methods. Representative methods include adaptive trade-off model (ATM) [18] and ensemble of constraint handling methods (ECHM) [19].

To address the second issue, the selection methods need to be designed to balance the performance of convergence and diversity in MOEAs. At present, MOEAs can be generally classified into three categories based on the selection strategies. They are Pareto-dominance (e.g., NSGA-II[20], PAES-II[21] and SPEA-II[22]), decomposition-based (e.g., MOEA/D [23], MOEA/D-DE[24], MOEA/D-M2M[25] and EAG-MOEA/D[26]) and indicator based methods (e.g., IBEA[27], R2-IBEA[28], SMS-EMOA[29] and HypE[30]). In the group of Pareto-dominance based methods, such as NSGA-II[20], the set of the first non-dominated level solutions is selected to improve the performance of convergence, and the crowding distance is adopted to maintain the performance of diversity. In the decomposition-based methods, the performance of convergence is maintained by minimizing the aggregation functions and the performance of diversity is obtained by setting the weight vectors uniformly. In the indicator based methods, such as HypE[30], the performance of convergence and diversity is achieved by the using the hypervolume metric.

A CMOP includes objectives and constraints. A number of features have already been identified to define the difficulty of objectives, which include:

  1. Geometry of PF (linear, convex, concave, degenerate, disconnected and mixed of them)

  2. Search space (biased, or unbiased)

  3. Unimodal or multi-modal objectives

  4. Dimensionality of variable space and objective space

The first one is the geometry of PF. The geometry of PF of a MOP can be linear, convex, concave, degenerate, disconnected and mixed of the them. Representative MOPs reflecting this type of difficulty include ZDT[31], F1-9 [32] and DTLZ [33]. The second one is the biased or unbiased search space. Representative MOPs in this category include MOP1-7 [34] and IMB1-14 [35]. The third one is the modality of objectives. The objectives of a MOP can be either uni-modal (DTLZ1 [33]) or multi-modal (F8 [32]). Objectives with multi-modal have multiple local optimal solutions which increase the likelihood of an algorithm being trapped in local optima. The high dimensionality of variable space and objective space are also critical features to define the difficulty of objectives. LSMOP1-9 [36] have high dimensionality in the variable space. DTLZ [33] and WFG [37] have high dimensionality in the objective space.

On the other hand, constraint functions in general greatly increase the difficulty of solving CMOPs. However, as far as we know, only several test suites (CTP[38], CF[39]) are designed for CMOPs.

CTP test problems [38] are have the capability of adjusting the difficulty of the constraint functions. They offer two types of difficulties: the difficulty near the Pareto front and the difficulty in the entire search space. The test problem CTP1 gives the difficulty near the PF, because the constraint functions of CTP1 make the search region close to the Pareto front infeasible. Test problems CTP2-CTP8 provide an optimizer the difficulty in the entire search space.

CF test problems [39] are also commonly used benchmarks, which provide two types of difficulties. For CF1-CF3 and CF8-CF10, their PFs are a part of their unconstrained PFs. The rest of CF test problems CF4-CF7 have difficulties near their PFs, and many constrained Pareto optimal points lie on some boundaries of the constraints.

Even though CDP [38] and CF [39] offer the above-mentioned advantages. They have some limitations:

  • The number of decision variables in the constraint functions can not be extended.

  • The difficulty level of each type is not adjustable.

  • No constraint functions with low ratios of feasible regions in the entire search space are suggested.

  • The number of objectives is not scalable.

Some other used two-objective test problems, include BNH [40], TNK [41], SRN [42] and OSY [43] problems, which are not scalable to the number of objectives, and difficult to identify types of difficulties.

In this paper, we propose a general framework to construct difficulty adjustable and objective scalable CMOPs which can overcome the limitations of existing CMOPs. CMOPs constructed by this toolkit can be classified into three major types, which are feasibility-hard, convergence-hard and diversity-hard CMOPs. Feasibility-hard CMOP is a type of problem that presents difficulty for CMOEAs to find feasible solutions in the search space. CMOPs with feasibility-hardness usually have small portions of feasible regions in the entire search space. In addition, CMOPs with convergence-hardness mainly suggest difficulty for CMOEAs to approach the PFs efficiently by setting many obstacles before the PFs, while CMOPs with diversity-hardness mainly provide difficulty for CMOEAs to distribute their solutions along the complete PFs. In our work, the three types of difficulty are embedded into the CMOPs through proper construction of constraint functions.

In summary, the contribution of this paper is as follows:

  1. This paper defines three primary types of difficulty for constraints in CMOPs. When designing new constraint handling mechanisms for a CMOEA, one has to investigate the nature of constraints in a CMOP that the CMOEA is aiming to address, including the types and levels of difficulties embedded in the constraints. Therefore, a proper definition on the types of difficulty for constraints in CMOPs is necessary and desirable.

  2. This paper also defines the level of difficulty, regarding each type of difficulty for constraints in the constructed CMOPs, which can be adjusted by users. A difficulty level is uniquely defined by a triplet with each of its parameter specifying the level of each primary difficulty type respectively. Combination of the three primary constraint types with different difficulty triplets can lead to construction of a large variety of constraints for CMOPs.

  3. Based on the proposed three primary types of difficulty for constraints, nine difficulty adjustable CMOPs named DAS-CMOP1-9 are constructed.

The remainder of this paper is organized as follows. Section II discusses the effects of constraints on PFs. Section III introduces the types and levels of difficulties provided by constraints in CMOPs. Section IV explains the proposed toolkit of construction methods for generating constraints in CMOPs with different types and levels of difficulty. Section V realizes the scalability to the number of objectives in CMOPs using the proposed toolkit. Section VI generates a set of difficulty adjustable CMOPs using the proposed toolkit. In Section VII, the performance of two CMOEAs on DAS-CMOP1-9 with different difficulty levels are compared by experimental studies, and Section VIII concludes the paper.

Ii Effects of constraints on PFs

Constraints define the infeasible regions in the search space, leading to different types and levels of difficulty for the resulting CMOPs. Some major effects of the constraints on PFs in CMOPs include the following [44]:

  1. Infeasible regions make the original unconstrained PF partially feasible. This can be further divided into two situations. In the first situation, the PF of the constrained problem consists of a part of its unconstrained PF and a set of solutions on some boundaries of constraints, as illustrated by Fig. 1(a). In the second situation, the PF of the constrained problem is only a part of its unconstrained PF, as illustrated by Fig. 1(b).

  2. Infeasible regions block the way towards the PF, as illustrated by Fig. 1(c).

  3. The complete original PF is covered by infeasible regions and becomes no more feasible. Every constrained Pareto optimal point lies on some constraint boundaries, as illustrated by Fig. 1(d).

  4. Constraints may reduce the dimensionality of the PF with one example illustrated by Fig. 1(e). In general, although the problem is dimensional, constraints make the constrained PF dimensional (where ). In the particular case of Fig. 1(e), .

(a)     (b)     (c)     (d)     (e)
Fig. 1: Illustration of the effects of constraints on PFs. (a) Infeasible regions makes the original unconstrained PF partially feasible. Many constrained Pareto optimal solutions lie on the constraint boundaries. (b) Infeasible regions makes the original unconstrained PF partially feasible. The constrained PF is a part of its unconstrained PF. (c) Infeasible regions blocks the way of converging to the PF. The constrained PF is same to its unconstrained PF. (d) The complete original PF is no more feasible. Every constrained Pareto optimal solution lies on the constraint boundaries. (e) Constraints reduce the dimensionality of the PF. A two-objective optimization problem is transformed into a constrained single optimization problem.

Iii Difficulty types and levels of CMOPs

Three primary difficulty types have been identified, including convergence-hardness, diversity-hardness, and feasibility-hardness. A difficulty level for each primary difficulty type can be defined as a parameter ranging from 0 to 1. Three difficulty levels, corresponding to three primary difficulty types respectively, form a triplet that depicts the nature of the difficulty of a CMOP.

Iii-a Difficulty 1: Diversity-hardness

Generally, the PFs of CMOPs with diversity-hardness have many discrete segments, or some parts more difficult to be achieved than the other parts by imposing large infeasible regions near them. As a result, achieving the complete PF is difficult for CMOPs.

Iii-B Difficulty 2: Feasibility-hardness

For the feasibility-hard CMOPs, the ratios of feasible regions in the search space are usually very low. It is difficult to generate a feasible solution for a CMOEA on the feasibility-hard CMOPs. Often in the initial stage of a CMOEA, most solutions in the population are infeasible.

Iii-C Difficulty 3: Convergence-hardness

CMOPs with convergence-hardness hinder the convergence of CMOEAs towards the PFs. Usually, CMOEAs encounter more difficulty to approach the PFs. Because infeasible regions block the way of CMOEAs converging to the PFs. In other words, the generational distance (GD) metric [45], which indicates the performance of convergence, is difficult to be minimized in the evolutionary process.

Iii-D Difficulty level of each primary difficulty type

A difficulty level of each primary difficulty type can be defined by a parameter in the parameterized constraint function corresponding to the primary difficulty type. Each parameter is normalized from 0 to 1. Three parameters, corresponding to the difficulty level of the three primary difficulty types respectively, form a triplet that exactly defines the nature of difficulty of a CMOP constructed by the three parameterized constraint functions.

If each element of the triplet can only take value of either 0 or 1, then a simple combination of the three primary difficulty types will give rise to seven basic different difficulty types. This is analogous to a simple combination of three primary colors gives rise to seven basic colors. But if we allow the three parameters to take any value between 0 and 1, then we can literally get countless difficulty nature (analogous to countless colors in the color space). A difficulty nature here is then precisely depicted by a triplet .

Fig. 2: The illustration of three primary difficulty types and their combination resulting in seven basic difficulty types (as shown in Table I), using an analogy of three primary colors and their combination towards seven basic colors.
Fig. 3: The illustration of combining three parameterized constraint functions using a triplet composing of three parameters. The three primary constraint functions correspond to the three primary difficulty types respectively.
Basic Difficulty Types Comment
T1: Diversity-hardness Distributing the feasible solutions in the complete PF is difficult.
T2: Feasibility-hardness Obtaining a feasible solution is difficult.
T3: Convergence-hardness Approaching a Pareto optimal solution is difficult.
T4: Diversity-hardness and feasibility-hardness Obtaining a feasible solution and the complete PF is difficult.
T5: Diversity-hardness and convergence-hardness Approaching a Pareto optimal solution and the complete PF is difficult.
T6: Feasibility-hardness and convergence-hardness Obtaining a feasible solution and approaching a Pareto optimal solution is difficult.
T7: Diversity-hardness, feasibility-hardness and convergence-hardness Obtaining a Pareto optimal solution and the complete PF is difficult.
TABLE I: Basic difficulty types of the CMOPs

Iv construction toolkit

As we know, constructing a CMOP is composed of constructing two major parts - objective functions and constraint functions. Li, et al. [46] suggested a general framework for constructing objective functions. It is stated as follows:

(2)

where are two sub-vectors of . The function is called the shape function, and is called the nonnegative distance function. The objective function is the sum of the shape function and the nonnegative distance function . We adopt Li, et al.’s method [46] in this work.

In terms of constructing the constraint functions, three different types of constraint functions are suggested in this paper, corresponding to the proposed three primary types of difficulty of CMOPs. More specifically, Type-I constraint functions provide the difficulty of diversity-hardness, Type-II constraint functions introduce the difficulty of feasibility-hardness, and Type-III constraint functions generate the difficulty of convergence-hardness. The detailed definition of the three types of constraint functions are given in detail as follows:

Iv-a Type-I Constraint Functions: Diversity-hardness

Type-I constraint functions are defined to limit the boundary of sub-vector . More specifically, this type of constraint functions divides the PF of a CMOP into a number of disconnected segments, generating the difficulty of diversity-hardness. Here, we use a parameter to represent the level of difficulty. means the constraint functions impose no effects on the CMOP, while means the constraint functions provide their maximum effects.

An example of CMOP with diversity-hardness is suggested as follows:

(3)

where , . As an example , are set here. The parameter indicating the level of difficulty is set to . The number of disconnected segments in the PF is controlled by . Moreover, the value of controls the width of each segment. The width of segments reaches its maximum when . When increases, the width of segments decreases, and the difficulty level increases, so does the parameter of the difficulty level . As a result, if is set to , the PF is shown in Fig. 4(a). If , the PF is shown in Fig.4 (b). It can be observed that the width of segments of the PF is reduced as keeps increasing. If , the width of segments shrinks to zero, which provides the maximum level of difficulty to the CMOP. The PF of a three-objective CMOP with Type-I constraint functions is also shown in Fig. 4(d), with the difficult level . It can be seen that Type-I constraint functions can be applied in more than two-objective CMOPs, which means that a CMOP with the scalability to the number of objectives can be constructed using this type of constraints.

(a) (b) (c) (d)
Fig. 4: Illustrations on the influence of Type-I constraint functions. When the parameter of difficulty level increases, the width of segments in the PF decreases, and the difficulty level of a CMOP increases. Because the PF of a CMOP with Type-I constraint is disconnect and usually has many discrete segments, obtaining the complete PF is difficult. Thus a CMOP with Type-I constraints is diversity-hard. (a) Two-objective CMOP with . (b) Two-objective CMOP with . (c) Two-objective CMOP with . (d) Three-objective CMOP with .

Iv-B Type-II Constraint Functions: Feasibility-hardness

Type-II constraint functions are set to limit the reachable boundary of the distance function of , and thereby control the ratio of feasible regions. As a result, Type-II constraint functions generate the difficulty of feasibility-hardness. Here, we use a parameter to represent the level of difficulty, which ranges from 0 to 1. means the constraints are the weakest, and means the constraint functions are the strongest.

For example, a CMOP with Type-II constraint functions can be defined as follows:

(4)

where equals to , and , and . The distance between the constrained PF and unconstrained PF is controlled by , and in this example. The ratio of feasible regions is controlled by . If , , the feasible area reaches maximum as shown in Fig. 5(a). If , , the feasible area is decreased as shown in Fig. 5(b). If , , the feasible area in the objective space is very small. The PF of this problem is shown in Fig. 5(c). Type-II constraints can be also applied to CMOPs with three objectives as shown in Fig. 5(d).

(a) (b) (c) (d)
Fig. 5: Illustrations on the influence of Type-II constraint functions. The parameter of difficulty degree . Here, the default value of is set to 0.5. The ratio of feasible regions is controlled by . When the parameter increases, the portion of feasible regions decreases, and the difficulty level of feasibility increases. (a) . (b) . (c) . (d) The Type-II constraint can be applied into three-objective optimization problems, and .

Iv-C Type-III Constraint Functions: Convergence-hardness

Type-III constraint functions limit the reachable boundary of objectives. As a result, infeasible regions act like ’blocking’ hindrance for searching populations of CMOEAs to approach the PF. As a result, Type-III constraint functions generate the difficulty of convergence-hardness. Here, we use a parameter to represent the level of difficulty, which ranges from 0 to 1. means the constraints are the weakest, means the constraints are the strongest, and the difficulty level increases as increases.

For example, a CMOP with Type-III constraint functions can be defined as follows:

(5)

where the level of difficulty parameter is defined as . If , the PF is shown in Fig. 6(a). If , the infeasible regions are increased and shown in Fig. 6(b). If , the infeasible regions become bigger than those of as shown in Fig. 6(c). The constraints of Type-III can be also applied to CMOPs with three objectives as shown in Fig. 6(d).

Type-III constraint functions can be expressed in a matrix form, which can be defined as follows:

(6)

where . is a translation vector. is a transformational matrix, which control the degree of rotation and stretching of the vector . According to the Type-III constraint functions in Eq. (5), , and can be expressed as follows:

It is worthwhile to point out that by using this approach we can further extend the number of objectives to be more than three, even though more sophisticated visualization approach is needed to show the resulting CMOPs in the objective space.

(a) (b) (c) (d)
Fig. 6: Illustrations on the influence of Type-III constraint functions. Infeasible regions block the way of converging to the PF. The gray parts of each figure are infeasible regions. A parameter is adopted to represent the level of difficulty, which ranges from 0 to 1. means the constraints are the weakest, and means the constraints are the strongest. When increases, the difficulty level of convergence-hardness of a CMOP increases. (a) . (b) . (c) . (d) The Type-III constraint can also be applied into three-objective optimization problems, here with .

To summarize, the three types of constraint functions discussed above correspond to the three primary difficulty types of CMOPs respectively. In particular, Type-I constraint function corresponds to diversity-hardness, Type-II corresponds to feasibility-hardness, and Type-III corresponds to convergence-hardness. The level of each primary difficulty type can be decided by a parameter. In this work, three parameters are defined in a triplet , which specifies the difficulty level of a particular difficulty type. It is noteworthy to point out that this approach of constructing toolkit for CMOPs can also be scaled to generate CMOPs with more than three objective functions. The scalability to the number of objectives is discussed in more detail in Section V.

V Scalability to the number of objectives

Recently many-objective optimization attracts a lot of research interests, which makes the feature of scalability to the number of objectives of CMOPs desirable. A general framework to construct CMOPs with the scalability to the number of objectives is given in Eq. (7).

In Eq. (7), we borrow the idea of WFG toolkit [37] to construct objectives, which can be scaled to any number of objectives. More specifically, the number of objectives is controlled by a user-defined parameter .

Three different types of constraint functions proposed in Section V can be combined together with the scalable objectives to construct difficulty adjustable and scalable CMOPs (DAS-CMOPs). More specifically, the first constraint functions with Type-I are defined to limit the reachable boundary of each decision variable in the shape functions ( to ), which have the ability to control the difficulty level of diversity-hardness by . The to constraint functions belong to Type-II, which limit the reachable boundary of the distance functions ( to ). They have the ability to control the the difficulty level of feasibility-hardness by . The last constraint functions are set directly on each objective, and belong to Type-III. They generate a number of infeasible regions, which hinder the working population of a CMOEA approaching to the PF. The difficulty level of convergence-hardness generated by Type-III constraint functions is controlled by . The rest of parameters in Eq. (7) are illustrated as follows.

Three parameters , and are used to control the number of each type of constraint functions, respectively. and . The total number of constraint functions is controlled by . decides the dimensions of decision variables, and . decides the number of disconnected segments in the PF. indicates the distance between the constrained PF and the unconstrained PF. The difficulty level of a DAS-CMOP is controlled by a difficulty triplet , with each of its component ranging from 0 to 1. When each of parameter in the difficult triplet increases, the difficulty level of a DAS-CMOP increases.

It is worth noting that the number of objectives of DAS-CMOPs can be easily scaled by tuning the parameter of . The difficulty level of DAS-CMOPs can be also easily adjusted by assigning a difficulty triplet with three parameters ranging from 0 to 1.

(7)

Vi A set of difficulty adjustable and scalable CMOPs

In this section, as an example, a set of nine difficulty adjustable and scalable CMOPs (DAS-CMOP1-9) is suggested through the proposed toolkit.

As mentioned in Section IV, constructing a CMOP composes of constructing objective functions and constraint functions. According to Eq. (7), we suggest nine multi-objective functions, including convex, concave and discrete PF shapes, to construct CMOPs. A set of difficulty adjustable constraint functions is generated by Eq. (7). Nine difficulty adjustable and scalable CMOPs named DAS-CMOP1-9 are generated by combining the suggested objective functions and the generated constraint functions. The detailed definitions of DAS-CMOP1-9 are shown in Table II.

In Table II, DAS-CMOP1-3 have the same constraint functions. For DAS-CMOP4-6, they also have the same constraint functions. The difference between DAS-CMOP1-3 and DAS-CMOP4-6 is that they have different distance functions. For DAS-CMOP1-6, they have two objectives. The number of objectives in Eq. (7) are able to scale to more than two. For example, DAS-CMOP7-9 have three objectives. The constraint functions of DAS-CMOP8 and DAS-CMOP9 are the same as those of DAS-CMOP7.

It is worth noting that the value of difficulty triplet elements can be set by users. If we want to reduce/increase the difficulty levels of DAS-CMOP1-9, we only need to adjust the parameters of the triplet elements to smaller/larger values, and generate a new set of test instances.

Problem Objectives Constraints
DAS-CMOP1
DAS-CMOP2 It is the same as that of DAS-CMOP1
DAS-CMOP3 It is the same as that of DAS-CMOP1
DAS-CMOP4
DAS-CMOP5 It is the same as that of DAS-CMOP4
DAS-CMOP6 It is the same as that of DAS-CMOP4
DAS-CMOP7
DAS-CMOP8 It is the same as that of DAS-CMOP7
DAS-CMOP9 It is the same as that of DAS-CMOP7
TABLE II: DAS-CMOPs Test suite: the objective functions and constraint functions of DAS-CMOP1-9.

Vii Experimental Study

Vii-a Experimental Settings

To test the performance of CMOEAs on the DAS-CMOPs, two commonly used CMOEAs (i.e., MOEA/D-CDP and NSGA-II-CDP) are tested on DAS-CMOP1-9 with sixteen different difficulty triplets in the experiment. As descripted in Section IV, three parameters are defined in a triplet , which specifies the difficulty level of a particular difficulty type. More specifically, represents the difficulty level of diversity-hardness, denotes the difficulty level of feasibility-hardness, and indicates the difficulty level of convergence-hardness. The difficulty triplets for each DAS-CMOP are listed in Table III.

Difficulty Triplets
(0.0,0.0,0.0) (0.0,0.5,0.0) (0.0,0.0,0.75) (1.0.0,0.0,0.0)
(0.0,0.25,0.0) (0.0,0.0,0.5) (0.75,0.0,0.0) (0.25,0.25,0.25)
(0.0,0.0,0.25) (0.5,0.0,0.0) (0.0,1.0,0.0) (0.5,0.5,0.5)
(0.25,0.0,0.0) (0.0,0.75,0.0) (0.0,0.0,1.0) (0.75,0.75,0.75)
TABLE III: The difficulty triplets for each DAS-CMOP

The detailed parameters of the algorithms are summarized as follows.

  1. Setting for reproduction operators: The mutation probability

    ( is the number of decision variables). For the polynomial mutation operator, the distribution index is set to 20. For the simulated binary crossover (SBX) operator, the distribution index is set to 20. The rate of crossover .

  2. Population size: For DAS-CMOP1-6, , and for DAS-CMOP7-9, .

  3. Number of runs and stopping condition: Each algorithm runs 30 times independently on each test problem with sixteen different difficulty triplets. The maximum function evaluations is 100000 for DAS-CMOP1-6, 200000 for DAS-CMOP7-9.

  4. Neighborhood size: for DAS-CMOP1-6, for DAS-CMOP7-9.

  5. Probability use to select in the neighborhood: .

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

Vii-B Performance Metric

To measure the performance of MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP1-9 with different difficulty triplets, the inverted generation distance ()[47] is adopted. The detailed definition of is given as follows:

  • Inverted Generational Distance ():

The metric simultaneously reflects the performance of convergence and diversity, and it is defined as follows:

(8)

where is the ideal PF set, is an approximate PF set achieved by an algorithm. represents the number of objectives. It is worth noting that the smaller value of represents the better performance of both diversity and convergence.

Vii-C Performance Comparisons on two-objective DAS-CMOPs

Table IV presents the statistic results of values for MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP1-3. We can observe that for DAS-CMOP1 with difficulty triplets , and , NSGA-II-CDP is significantly better than MOEA/D-CDP, which indicates that NSGA-II-CDP is more suitable for solving DAS-CMOP1 with feasibility-hardness. For DAS-CMOP1 with difficulty triplets , and , MOEA/D-CDP is significantly better than NSGA-II-CDP, which indicates that MOEA/D-CDP is more suitable for solving DAS-CMOP1 with convergence-hardness. For DAS-CMOP1 with simultaneous convergence-, feasibility- and convergence-hardness, i.e., the difficulty triplets are , and , NSGA-II-CDP is significantly better than MOEA/D-CDP.

The final populations with the best values in 30 independent runs by using MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP1 with difficulty triplets , , , and are plotted in Fig. 7. We can observe that each type of constraint functions in DAS-CMOP1 indeed generates corresponding difficulties for MOEA/D-CDP and NSGA-II-CDP. With the increasing of each elements in the difficulty triplet, the problem is more difficult to solve, as illustrated by Fig. 7(d)-(e) and Fig. 7(i)-(j).

For DAS-CMOP2 with feasibility-hardness, for example, the difficulty triplets are , and , NSGA-II-CDP is significantly better than MOEA/D-CDP. For DAS-CMOP2 with difficulty triplets and , MOEA/D-CDP is significantly better than NSGA-II-CDP, which indicates that MOEA/D-CDP is more suitable for solving DAS-CMOP2 with convergence-hardness. For DAS-CMOP2 with the difficulty triplet , MOEA/D-CDP is also significantly better than NSGA-II-CDP. For DAS-CMOP2 with simultaneous convergence-, feasibility- and convergence-hardness, NSGA-II-CDP is significantly better than MOEA/D-CDP.

Fig. 8 shows the final populations with the best values in 30 independent runs by using MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP2 with difficulty triplets , , , and . Both MOEA/D-CDP and NSGA-II-CDP can only achieve a few parts of the PFs. With the increasing of each element in the difficulty triplet, it is more difficult for MOEA/D-CDP and NSGA-II-CDP to find the whole PFs of DAS-CMOP2.

For DAS-CMOP3 with the difficulty triplet , that is, there are no constraints in DAS-CMOP3, NSGA-II-CDP is significantly better than MOEA/D-CDP. For DAS-CMOP3 with difficulty triplets and , MOEA/D-CDP is significantly better than NSGA-II-CDP, which indicates that MOEA/D-CDP is more suitable for solving DAS-CMOP3 with larger difficulty levels in terms of convergence-hardness. For DAS-CMOP3 with the difficulty triplet , MOEA/D-CDP performs better than NSGA-II-CDP. For DAS-CMOP3 with feasibility-hardness, for example, the difficulty triplets are , and , NSGA-II-CDP is significantly better than MOEA/D-CDP. For DAS-CMOP3 with simultaneous convergence-, feasibility- and convergence-hardness, NSGA-II-CDP is significantly better than MOEA/D-CDP.

Fig. 9 shows the final populations with the best values in 30 independent runs by using MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP3 with difficulty triplets , , , and . For DAS-CMOP3 with diversity- or feasibility- or convergence-hardness, MOEA/D-CDP and NSGA-II-CDP can not find the whole PFs. With the increasing of difficulty triplets, DAS-CMOP3 is becoming more difficult for MOEA/D-CDP and NSGA-II-CDP to solve, as illustrated by Fig. 9(d)-(e) and Fig. 9(i)-(j).

The statistic results of values for MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP4-7 are shown in Table V. From this Table, we can observe that MOEA/D-CDP is significantly better than MOEA/D-CDP on DAS-CMOP4 with the difficulty triplet . In other words, MOEA/D-CDP works better than NSGA-II-CDP on DAS-CMOP4 without any constraints. For DAS-CMOP4 with feasibility-hard difficulty triplets , and , NSGA-II-CDP is significantly better than MOEA/D-CDP.

For DAS-CMOP4 with diversity-hard difficulty triplets , , and , MOEA/D-CDP performs significantly better than NSGA-II-CDP. For DAS-CMOP4 with convergence-hard difficulty triplets , and , MOEA/D-CDP also performs significantly better than NSGA-II-CDP.

For DAS-CMOP5 with feasibility-hard difficulty triplets , , and , NSGA-II-CDP performs significantly better than MOEA/D-CDP. For DAS-CMOP5 with diversity- or convergence-hardness, MOEA/D-CDP is significantly better than NSGA-II-CDP.

For DAS-CMOP6 with the difficulty triplet , NSGA-II-CDP is significantly better than MOEA/D-CDP. For DAS-CMOP6 with difficulty triplets , , and , MOEA/D-CDP is significantly better than NSGA-II-CDP. For DAS-CMOP5 with the rest of difficulty triplets, MOEA/D-CDP and NSGA-II-CDP have not any significantly difference.

Vii-D Performance Comparisons on three-objective DAS-CMOPs

The statistic results of values for MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP7-9 are presented in Table IV. For DAS-CMOP7 without any constraints, that is, the difficulty triplet is , NSGA-II-CDP is significantly better than MOEA/D-CDP. For DAS-CMOP7 with diversity- or feasibility-hardness, for example, the difficulty triplets are , , , and , NSGA-II-CDP is also significantly better than MOEA/D-CDP. For DAS-CMOP7 with difficulty triplets and , MOEA/D-CDP is significantly better than NSGA-II-CDP. For DAS-CMOP7 with simultaneous diversity-, feasibility- and convergence-hardness, NSGA-II-CDP performs significantly better than NSGA-II-CDP.

For DAS-CMOP8 with convergence-hardness, for example, the difficulty triplets are , , and , MOEA/D-CDP performs significantly better than NSGA-II-CDP. For DAS-CMOP8 with the difficulty triplet , MOEA/D-CDP is also significantly better than NSGA-II-CDP. For DAS-CMOP8 with the rest of difficulty triplets, NSGA-II-CDP is better or significantly better than MOEA/D-CDP.

For DAS-CMOP9 with feasibility-hardness, i.e, the difficulty triplets are , and , NSGA-II-CDP performs significantly better than MOEA/D-CDP. For DAS-CMOP9 with convergence- or diversity-hardness, i.e., the difficulty triplets are , , , , , MOEA/D-CDP is significantly better than NSGA-II-CDP. For DAS-CMOP9 with simultaneous diversity-, feasibility- and convergence-hardness, NSGA-II-CDP performs significantly better than MOEA/D-CDP. The final populations with the best values in 30 independent runs by using MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP9 with difficulty triplets , , , and are plotted in Fig. 10. We can observe that both MOEA/D-CDP and NSGA-II-CDP only achieve a few parts of PFs of DAS-CMOP9.

Vii-E Analysis of Experimental Results

From the above performance comparisons on the nine test instances DAS-CMOPs, it is clear that each type of constraint functions generates corresponding difficulties for MOEA/D-CDP and NSGA-II-CDP. With the increasing of each elements in the difficulty triplet, the problem is becoming more difficult for MOEA/D-CDP and NSGA-II-CDP to solve. Furthermore, it can be concluded that NSGA-II-CDP performs better than MOEA/D-CDP on DAS-CMOPs with feasibility-hardness. MOEA/D-CDP performs better than NSGA-II-CDP on DAS-CMOPs with diversity- or convergence-hardness. In the case of DAS-CMOPs with simultaneous diversity-, feasibility- and convergence-hardness, NSGA-II-CDP performs better than MOEA/D-CDP on most of test instances.

Instance DAS-CMOP1 DAS-CMOP2 DAS-CMOP3
Difficulty Triplet MOEA/D-CDP NSGA-II-CDP MOEA/D-CDP NSGA-II-CDP MOEA/D-CDP NSGA-II-CDP
(0.0,0.0,0.0) 1.367E-01 1.494E-01 1.678E-01 1.720E-01 1.909E-01 1.536E-01
2.934E-02 3.723E-02 2.675E-02 3.710E-02 4.315E-02 2.340E-02
(0.0,0.25,0.0) 1.375E-01 9.878E-02 1.203E-01 7.994E-02 1.744E-01 1.224E-01
2.591E-02 1.185E-02 2.097E-02 1.316E-02 4.493E-02 2.374E-02
(0.0,0.0,0.25) 1.351E-01 2.006E-01 1.709E-01 1.606E-01 2.076E-01 1.888E-01
3.515E-02 2.100E-02 3.485E-02 4.088E-02 4.827E-02 3.296E-02
(0.25,0.0,0.0) 1.497E-01 1.524E-01 1.719E-01 1.796E-01 2.300E-01 2.146E-01
2.609E-02 4.081E-02 3.073E-02 4.273E-02 5.525E-02 5.225E-02
(0.0,0.5,0.0) 1.578E-01 1.174E-01 1.359E-01 9.157E-02 1.847E-01 1.292E-01
1.516E-02 1.304E-02 2.820E-02 1.670E-02 3.752E-02 2.305E-02
(0.0,0.0,0.5) 1.809E-01 2.362E-01 1.580E-01 2.164E-01 2.127E-01 2.095E-01
3.005E-02 2.125E-02 4.670E-02 4.156E-02 4.019E-02 1.883E-02
(0.5,0.0,0.0) 1.602E-01 1.605E-01 1.796E-01 1.799E-01 4.185E-01 3.263E-01
4.631E-02 3.716E-02 3.874E-02 3.604E-02 1.068E-01 1.168E-01
(0.0,0.75,0.0) 1.858E-01 1.420E-01 1.548E-01 1.073E-01 2.320E-01 1.527E-01
3.527E-02 1.546E-02 2.665E-02 1.557E-02 4.604E-02 2.729E-02
(0.0,0.0,0.75) 1.769E-01 3.147E-01 1.554E-01 3.354E-01 2.201E-01 2.656E-01
3.770E-02 5.324E-02 2.601E-02 1.110E-01 2.596E-02 5.844E-02
(0.75,0.0,0.0) 1.658E-01 1.510E-01 2.206E-01 1.712E-01 2.258E-01 1.956E-01
4.960E-02 4.018E-02 4.483E-02 4.269E-02 5.725E-02 5.785E-02
(0.0,1.0,0.0) 3.682E-01 3.636E-01 3.250E-01 3.235E-01 4.353E-01 4.389E-01
1.230E-02 6.217E-03 7.175E-03 4.915E-03 4.106E-02 2.952E-02
(0.0,0.0,1.0) 6.998E-01 4.625E-01 7.388E-01 7.177E-01 6.590E-01 6.615E-01
3.767E-01 2.755E-02 1.901E-01 1.254E-01 2.879E-04 1.982E-03
(1.0.0,0.0,0.0) 4.515E-01 1.531E+00 4.102E-01 1.413E+00 4.331E-01 1.842E+00
1.169E-01 9.908E-01 1.029E-01 8.854E-01 9.567E-02 1.431E+00
(0.25,0.25,0.25) 2.674E-01 2.119E-01 1.404E-01 9.860E-02 2.589E-01 1.686E-01
3.893E-02 4.025E-02 2.999E-02 1.632E-02 3.644E-02 4.206E-02
(0.5,0.5,0.5) 4.160E-01 3.624E-01 1.656E-01 1.123E-01 4.680E-01 4.027E-01
7.441E-02 4.702E-02 4.357E-02 1.638E-02 5.516E-02 3.817E-02
(0.75,0.75,0.75) 7.750E-01 7.079E-01 1.867E-01 1.029E-01 7.473E-01 2.808E-01
5.452E-02 9.615E-02 3.758E-02 1.201E-02 1.864E-01 1.242E-01
TABLE IV:

Mean and standard deviation of

values obtained by MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP1-3. Wilcoxon’s rank sum test at 0.05 significance level is performed between MOEA/D-CDP and NSGA-II-CDP. and denote that the performance of NSGA-II-CDP is significantly worse than or better than that of MOEA/D-CDP, respectively.
Instance DAS-CMOP4 DAS-CMOP5 DAS-CMOP6
Difficulty Triplet MOEA/D-CDP NSGA-II-CDP MOEA/D-CDP NSGA-II-CDP MOEA/D-CDP NSGA-II-CDP
(0.0,0.0,0.0) 1.121E-02 5.282E-02 1.464E-02 5.356E-02 6.304E-02 6.778E-02
8.906E-03 2.977E-02 1.365E-02 2.640E-02 7.337E-02 3.402E-02
(0.0,0.25,0.0) 2.913E-03 2.412E-03 3.034E-03 2.269E-03 6.410E-02 5.489E-02
1.290E-03 1.460E-04 1.265E-03 7.276E-05 6.666E-02 5.249E-02
(0.0,0.0,0.25) 7.880E-02 3.380E-01 4.158E-02 4.192E-01 1.202E-01 3.907E-01
4.741E-02 5.514E-02 2.298E-02 1.665E-01 5.496E-02 2.475E-01
(0.25,0.0,0.0) 1.849E-02 6.475E-02 2.010E-02 6.510E-02 9.321E-02 1.291E-01
1.324E-02 3.587E-02 1.235E-02 4.145E-02 7.435E-02 6.290E-02
(0.0,0.5,0.0) 2.915E-03 2.345E-03 2.940E-03 2.390E-03 7.189E-02 5.139E-02
1.108E-03 7.926E-05 1.187E-03 6.758E-04 4.962E-02 5.530E-02
(0.0,0.0,0.5) 2.380E-01 9.456E-01 1.010E-01 8.761E-01 4.169E-01 9.448E-01
8.551E-02 3.330E-01 7.600E-02 2.850E-01 3.895E-01 3.083E-01
(0.5,0.0,0.0) 2.321E-02 5.856E-02 1.822E-02 5.840E-02 9.403E-02 1.369E-01
1.611E-02 3.015E-02 9.361E-03 2.992E-02 5.364E-02 5.124E-02
(0.0,0.75,0.0) 2.668E-03 3.105E-03 2.587E-03 3.078E-03 6.957E-02 6.104E-02
7.843E-04 3.840E-03 1.237E-03 4.450E-03 7.664E-02 5.444E-02
(0.0,0.0,0.75) 4.260E-01 1.266E+00 5.002E-01 1.223E+00 1.260E+00 1.258E+00
1.443E-01 1.709E-01 1.142E-01 2.086E-01 3.391E-01 2.116E-01
(0.75,0.0,0.0) 1.690E-02 5.761E-02 1.992E-02 7.616E-02 1.446E-01 1.268E-01
1.301E-02 2.894E-02 1.608E-02 3.299E-02 1.056E-01 6.071E-02
(0.0,1.0,0.0) 5.229E-02 1.019E-01 1.178E-02 1.114E-01 1.816E-01 1.616E-01
1.307E-01 1.272E-01 3.037E-02 1.017E-01 1.024E-01 1.222E-01
(0.0,0.0,1.0) 1.824E+00 1.633E+00 1.946E+00 1.612E+00 1.983E+00 1.656E+00
9.650E-02 2.638E-01 3.058E-01 2.741E-01 1.490E-01 2.554E-01
(1.0.0,0.0,0.0) 4.795E-01 7.564E+00 4.465E-01 7.917E+00 5.126E-01 8.314E+00
1.055E-01 4.254E+00 9.627E-02 7.613E+00 9.984E-02 6.611E+00
(0.25,0.25,0.25) 3.392E-03 5.863E-02 2.754E-03 4.428E-02 1.719E-01 1.418E-01
1.941E-03 1.292E-01 6.112E-04 1.206E-01 1.534E-01 1.734E-01
(0.5,0.5,0.5) 1.219E-02 3.658E-01 5.861E-01 2.919E-01 7.863E-01 5.457E-01
3.938E-02 1.246E-01 7.668E-01 4.291E-01 5.005E-01 3.147E-01
(0.75,0.75,0.75) 2.341E-01 2.449E-01 1.319E+00 4.312E-01 1.014E+00 9.106E-01
1.670E-03 5.065E-02 4.447E-01 5.137E-01 3.383E-01 4.101E-01
TABLE V: Mean and standard deviation of values obtained by MOEA/D-CDP and NSGA-II-CDP on DAS-CMOP4-6. Wilcoxon’s rank sum test at 0.05 significance level is performed between MOEA/D-CDP and NSGA-II-CDP. and denote that the performance of NSGA-II-CDP is significantly worse than or better than that of MOEA/D-CDP, respectively.
Instance DAS-CMOP7 DAS-CMOP8 DAS-CMOP9
Difficulty Triplet MOEA/D-CDP NSGA-II-CDP MOEA/D-CDP NSGA-II-CDP MOEA/D-CDP NSGA-II-CDP
(0.0,0.0,0.0) 6.052E-02 5.150E-02 6.679E-02 6.679E-02 2.737E-01 3.158E-01
1.018E-03 4.413E-03 5.616E-04 5.958E-03 2.447E-01 1.784E-01
(0.0,0.25,0.0) 5.154E-02 4.946E-02 6.759E-02 6.453E-02 3.499E-01 2.084E-01
1.787E-03 2.179E-03 2.889E-03 1.798E-03 1.447E-01 1.231E-01
(0.0,0.0,0.25) 5.048E-02 5.288E-02 6.399E-02 7.098E-02 9.470E-02 3.315E-01
9.856E-04 5.451E-03 1.117E-03 3.921E-03 6.021E-02 1.805E-01
(0.25,0.0,0.0) 5.852E-02 4.869E-02 6.756E-02 6.326E-02 3.127E-01 3.458E-01
7.656E-04 4.057E-03 1.122E-03 3.801E-03 2.813E-01 1.977E-01
(0.0,0.5,0.0) 5.178E-02 4.891E-02 6.718E-02 6.455E-02 3.552E-01 2.347E-01
3.099E-03 1.839E-03 2.691E-03 2.882E-03 1.265E-01 1.042E-01
(0.0,0.0,0.5) 4.583E-02 6.307E-02 6.502E-02 7.726E-02 8.891E-02 3.847E-01
2.608E-04 1.707E-02 1.267E-03 7.831E-03 2.556E-02 1.532E-01