A novel adaptive control strategy for decomposition-based multiobjective algorithm

Computers & Operations Research 78 (2017) 94–107

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Computers & Operations Research journal homepage: www.elsevier.com/locate/caor

A novel adaptive control strategy for decomposition-based multiobjective algorithm Qiuzhen Lin, Chaoyu Tang, Yueping Ma, Zhihua Du, Jianqiang Li, Jianyong Chen n, Zhong Ming College of Computer Science and Software Engineering, Shenzhen University, Shenzhen 518060, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 7 October 2015 Received in revised form 11 April 2016 Accepted 23 August 2016 Available online 24 August 2016

Recently, evolutionary algorithm based on decomposition (MOEA/D) has been found to be very effective and efficient for solving complicated multiobjective optimization problems (MOPs). However, the selected differential evolution (DE) strategies and their parameter settings impact a lot on the performance of MOEA/D when tackling various kinds of MOPs. Therefore, in this paper, a novel adaptive control strategy is designed for a recently proposed MOEA/D with stable matching model, in which multiple DE strategies coupled with the parameter settings are adaptively conducted at different evolutionary stages and thus their advantages can be combined to further enhance the performance. By exploiting the historically successful experience, an execution probability is learned for each DE strategy to perform adaptive adjustment on the candidate solutions. The proposed adaptive strategies on operator selection and parameter settings are aimed at improving both of the convergence speed and population diversity, which are validated by our numerous experiments. When compared with several variants of MOEA/D such as MOEA/D, MOEA/D-DE, MOEA/D-DE þPSO, ENS-MOEA/D, MOEA/D-FRRMAB and MOEA/D-STM, our algorithm performs better on most of test problems. & 2016 Published by Elsevier Ltd.

Keywords: Evolutionary algorithm Multiobjective optimization Adaptive differential evolution Decomposition

fi (x ) ≤ fi (y ) for each i ∈ {1, 2, ... , m} and f j (x ) < f j (y ) for at least

1. Introduction In real-world engineering applications, there has been a growing interest in the field of multiobjective optimization, the aim of which is to optimize multiple objectives simultaneously [1– 3]. For example, the goals in job shop scheduling need to minimize the makespan, total workload, and critical workload, while the targets in product design have to reduce the cost of product and optimize its quality. Generally, multiobjective optimization problems (MOPs) can be stated as follows.

Minimiz F (x) = (f1(x), f2 (x), ... , fm (x)) Subject to: x ∈ Ω

(1) T

where x = (x1, x2, ... , x n) is a candidate solution and F: Ω → Rm consists of m objective functions, Ω and Rm are respectively the ndimensional decision space and m-dimensional objective space. As MOPs may have multiple naturally conflicting objectives, no single solution can guarantee the simultaneous optimality of all the objectives. Therefore, the Pareto domination relationship is an important criterion to evaluate the quality of solutions [4]. Suppose that x is said to dominate y (denoted as x≻y ) if and only if n

Corresponding author. E-mail addresses: [email protected] (Q. Lin), [email protected] (J. Chen).

http://dx.doi.org/10.1016/j.cor.2016.08.012 0305-0548/& 2016 Published by Elsevier Ltd.

one index j ∈ {1, 2, ... , m}. A solution x′ ∈ Ω is called Pareto-optimal or nondominated solution if and only if there does not exist another solution x ∈ Ω such that x≻x′. All the Pareto-optimal vectors compose Pareto-optimal set (PS) and their corresponding objective vectors are called Pareto-optimal front (PF). One important job of MOPs is to seek a set of nondominated solutions that are close to the true PF and distributed uniformly along the true PF, which can be provided to the decision maker as the alternative solutions. Recently, decomposition approach is embedded into evolutionary algorithms for tackling MOPs [5–7]. It is based on the facts that a Pareto-optimal solution for MOPs, under some mild conditions, could be an optimal solution of a scalar optimization problem, whose optimization target is an aggregation of all the objectives. Therefore, the finding of PF can be decomposed into a set of scalar problems [8]. MOEA/D [8] may be the first multiobjective evolutionary algorithm (MOEA) based on decomposition, where MOPs are transformed into a set of scalar subproblems and then each subproblem is optimized using the information from its several neighboring subproblems. After the report of MOEA/D, it has greatly attracted the interest of scientific researchers and thus numbers of MOEA/D variants have been designed to further improve its performance. Based on their improvements on different

Q. Lin et al. / Computers & Operations Research 78 (2017) 94–107

components of MOEA/D, most of the MOEA/D variants can be classified into the following three categories. The first kind proposes the improved generation strategies for weight vectors used in decomposition approach of MOEA/D. Both of UMOEA/D [9] and MOEA/D-UDM [10] present the uniform methods to build the aggregation coefficient vectors for the subproblems, which can produce an arbitrary number of weight vectors to fit the population size. The distributions of coefficient vectors are more uniform over the design space. However, it is pointed out in [11] that the uniformly distributed weight vectors are not guaranteed to produce uniformly distributed solutions. To tackle this problem, an approach of generating uniformly distributed search directions is presented accordingly [11]. As the above fixed weight vectors may not work very well for various MOPs with complicated PF, the adaptive adjustment strategies for weight vectors are respectively introduced in paλ -MOEA/D [12] and MOEA/D-AWA [13], both of which change the number of weight vectors based on the geometrical information of PF. Moreover, a new initialization method for weight vectors is also proposed in MOEA/D-AWA to generate the more uniform weight vectors on the hyperplane. The second type designs the enhanced evolutionary operators in MOEA/D. In [14], MOEA/D-DE is presented by substituting the simulated binary crossover with differential evolution (DE), where the maximal number of solutions replaced by a child solution is limited to preserve the population diversity. Making full use of the information from the neighbors and entire population, MOEA/DDE concurrently performs the local and global search in each generation. Four DE strategies are combined in MOEA/D-FRRMAB [5] as the evolutionary operators and an adaptive control strategy is designed to determine their application rates in an online manner. Moreover, a sliding window is used to record the recent fitness improvement rates achieved by the operators and a decaying mechanism is employed to increase the selection probability of the best operator. Similar with MOEA/D-FRRMAB, three DE strategies are adopted in ADEMO/D [15] to constitute the evolutionary operator pool, where the probability of each selected strategy is determined by its empirically estimated quality. Four different credit assignment techniques are cooperated with two selection methods, i.e., probability matching and adaptive pursuit, to update the selection probability in ADEMO/D. The third class presents the improved selection strategies for MOEA/D. In ENS-MOEA/D [16], an adaptive selection strategy for the neighborhood sizes (NSs) is proposed to alleviate the influence of NSs on the performance of MOEA/D. Different NSs are adjusted adaptively based on the former successful experiences and it is experimentally validated that different MOPs may prefer different settings of NSs. A stable matching (STM) model is reported in MOEA/D-STM [6] for the selection procedure of MOEA/D, where each subproblem is uniquely matched with a single potential solution based on their preference information. The convergence and population diversity during the evolutionary process are well balanced by the STM model and the experiments validate the superiority of MOEA/D-STM over the above MOEA/D variants, such as ENS-MOEA/D and MOEA/D-FRRMAB. Following the similar research direction on the match of candidate solutions and subproblems, an inter-relationship model is built based on their mutual-preferences [17]. Different from the STM model trying to tradeoff the convergence and diversity, it is essentially a diversity first and convergence second strategy, which enables the elitist solutions to explore the entire PF. Moreover, dominance-based selection approach is also investigated to combine with decomposition-based approach in MOEA/DD [18], which is able to balance the convergence and diversity when solving many-objective optimization problems. A systematic approach is accordingly proposed to generate widely spread weight vectors for a high-

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dimensional objective space and a mating restriction scheme is designed to fully exploit the mating parents chosen from the neighboring subregions. Moreover, other bio-inspired heuristic algorithms are also investigated to mix with MOEA/D for providing the improved performance, such as MOEA/D-DE þ PSO [19] and MOEA/D-ACO [20]. In the above-mentioned MOEA/D variants, it is found that most of them adopt a single DE strategy as the evolutionary operator due to its easy implementation and strong exploratory capability in complicated decision space. However, this may lead to some difficulties when tackling certain kinds of MOPs as different DE strategies behave very differently on various MOPs. One promising solution for this problem is to hybridize multiple DE strategies in MOEA/D, the advantages of which can be integrated to enhance the comprehensive search capability. Such experimental studies have been conducted in MOEA/D-FRRMAB and ADEMO/D, where the application rates of multiple DE strategies are dynamically determined. Working along the similar research direction, this paper proposes a novel adaptive DE selection strategy coupled with the adaptive parameter settings, which provides the enhanced optimization performance. Different from the adaptive strategies reported in MOEA/D-FRRMAB and ADEMO/D, our adaptive scheme is more promising as it not only adaptively chooses the DE strategies, but also automatically adjusts the associated parameter settings by exploiting the information of successful and failure solutions. By implementing our adaptive approach in MOEA/D-STM [6], a novel adaptive MOEA/D (AMOEA/D) is accordingly presented, which owns the stronger exploratory capability and better diversity. The performance of AMOEA/D is validated by the complicated UF benchmark problems [21]. The experimental results show that AMOEA/D outperforms some competitive variants of MOEA/D, such as MOEA/D [8], MOEA/D-DE [14], MOEA/D-DE þ PSO [19], ENS-MOEA/D [16], MOEA/D-FRRMAB [5] and MOEA/D-STM [6], on most of UF test problems. The remainder of this paper is organized as follows. Section 2 introduces the relevant backgrounds, including the decomposition approach, classical DE strategies, adaptive DE strategies and dynamic resource assignment in MOEA/D-DRA [22]. The details of our proposed algorithm AMOEA/D are presented in Section 3, where the proposed adaptive control strategies on operator selection and parameter settings are introduced. Experimental results of our algorithm with several MOEA/D variants are compared and analyzed in Section 4. At last, Section 5 summarizes our conclusions and gives our future work.

2. Backgrounds 2.1. Decomposition approach Currently, there are several decomposition approaches that can be used to transform MOPs into a number of aggregation subproblems, such as the weighted sum approach, Tchebycheff approach and boundary intersection method [23]. In this paper, Tchebycheff approach is utilized as it is mostly used in many variants of MOEA/D [6,8,14,24], which is expressed as follows.

min g tch x|w , z* = max x∈Ω

(

)

1≤ i ≤ m

{ f ( x) − z * /w } i

i

i

(2)

* ) is the vector of reference points, i.e., where z* = (z1*, z2*, ... , zm zi* = min {fi (x ) x ∈ Ω} for each i = 1, 2, ... , m. It is noted that when wi is set to 0, it will be replaced by wi = 10−6 in Eq. (2) to ensure that the division is valid. For each Pareto optimal point x*, there exists a weight vector w to make sure that x* is also the optimal solution of Eq. (2). Therefore, each optimal solution of Eq. (2) is also a Pareto optimal solution of Eq. (1) and a number of Pareto

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optimal solutions can be obtained by using a set of different weight vectors. 2.2. Classical differential evolution Differential evolution is one of the most successful and popular evolutionary algorithms to provide the strong global optimization ability, which owns the special features such as easiness of use, simple structure, efficiency and robustness [25,26]. With the rapid development of DE, it has been successfully applied for handling constrained, unconstrained, large scale, dynamic and uncertain MOPs that are modeled from the various scientific and engineering fields [27–29]. The first DE mutation strategy named DE/rand/1 is proposed in [30], which is applied for global optimization. After that, numbers of classical DE mutation strategies have been presented, including DE/rand/2, DE/best/1, DE/best/2, DE/current-torand/1, DE/current-to-rand/2 and DE/current-to-best/1 [30–32]. It is noted that these classical DE mutation strategies have their own advantages when solving various optimization problems. For example, as DE/best/1, DE/best/2, DE/rand-to-best/1 and DE/currentto-best/1 exploit the best solution found so far to do further exploration, they have a fast convergence speed and very good optimization performance when solving unimodal problems. However, in tackling multimodal problems, they are easy to get trapped and lead to premature convergence. On the other hand, DE/rand/1 and DE/rand/2 have a slow convergence speed but strong exploration capability to prevent premature [33], which are more suitable for multimodal problems. DE/current-to-rand/1 and DE/ current-to-rand/2 are essentially rotation-invariant, which perform better than the other DE strategies when tackling the rotated problems [32]. Obviously, different DE strategies have distinct search characteristics, which behave pretty differently in locating the optimal solution. 2.3. Adaptive differential evolution The distinct search capabilities of different DE strategies indicate that the optimization performance to a specific problem is extremely sensitive to the selected DE strategies and their control parameter settings. Generally, a trial-and-error search for the appropriate combination of DE strategies and their control parameter settings is necessarily required in order to provide better optimization performance. However, this procedure is very timeconsuming and the user may not understand the approach to tune the control parameters. To overcome the above disadvantages, a number of adaptive control schemes for DE have been well studied recently and it is experimentally confirmed that the adaptive control strategies coupled with a set of well-designed parameters can obviously enhance the overall optimization performance. This is usually achieved by dynamically selecting the DE strategies and adapting the associated parameters according to the characteristics of problem landscapes. Generally, the adaptive control schemes for DE are designed following the two research directions, i.e., adaptive selection of DE strategies and adaptive control of parameter settings, which are briefly introduced as follows. On the adaptive selection of DE strategies, probability matching (PM) and adaptive pursuit (AP) have been presented to adaptively select the DE strategies. In PM-ADAPSS-DE [34] and AdapSS-JADE [35], the PM method is embedded into DE to implement the adaptive selection strategy, where four credit assignment techniques based on the relative fitness improvement are compared experimentally. Moreover, the performance of PM and AP is also compared in AdapSS-JADE when they are combined with four DE strategies, and the simulation results show that the AP approach is better than the PM method to cooperate with DE. In SspDE [36], each target individual has an associated strategy list (SL). During

the evolution, a trial individual is generated by using a DE strategy taken from its SL. The selected DE strategy will enter a winning strategy list (wSL) when the generated trial individual is better than the original one. By this way, the best DE strategy will have the bigger selection probability for the next generation. Unlike the above adaptive selection of classical DE strategies, Sa-JADE [37] presents a family of improved DE variants, where the strategy adaptation mechanisms (SaM) are designed to adaptively pick a more suitable strategy for a specific problem. The proposed SaM are controlled by a strategy parameter, which is adaptively adjusted using the idea of parameter adaption in JADE [38] or the parameter self-adaptation in [39]. About the adaptive control of parameter settings, the main control parameters in DE include the crossover rate CR and scaling factor F , which greatly affect the optimization performance. For example, a small value of CR is usually preferred when solving separable problems, while a large one is suitable for tackling nonseparable problems [40]; a large value of F is generally demanded at the early stage of the evolution in order to maintain the population diversity, while a small one is able to speed up the convergence at the later stage. Therefore, there doesnot exist any fixed parameter setting that always performs very well for all types of optimization problems [41]. The adaptive control of parameter settings can well handle the above problems and find a suitable parameter setting for the target problem. Based on the control mechanisms on parameter settings, the adaptive control strategies can be generally classified into three categories [42]. The first kind is deterministic parameter control [43,44], where the parameter adaption takes place when the control parameters are altered by some deterministic rules without considering any feedback information from the evolutionary search. The second class is adaptive parameter control [38,45], which exploits some beneficial information from the feedback of the evolutionary search to dynamically decide the direction and/or the magnitude of the change to the parameters. The last one is self-adaptive parameter control [39,46], which utilizes the idea of “the evolution of evolution” to conduct the self-adaptation of control parameters. In this paper, as we adopt decomposition approach to transform MOPs into a set of aggregation subproblems, the abovementioned adaptive control mechanisms can be slightly modified to tackle MOPs. Inspired by the reported adaptive control strategies, a novel adaptive DE strategy is designed here for MOPs, in which the adaptive selection of DE strategies and adaptive control of parameter settings are combined. Based on historical experience of successful and failure solutions, our algorithm can adaptively adjust the probabilities for the selected DE strategies and their corresponding parameter settings in order to enhance the comprehensive optimization performance. 2.4. Dynamic resource assignment in MOEA/D Assume that N evenly spread weight vectors w = {w1, w 2, ... , w N} are available, where each m w i = {w1i, w2i , ... , wmi} (i ∈ [1, N ]) satisfies ∑k = 1 wki = 1 and wki ≥ 0 for all k ∈ [1, m]. Then, the approximation of PF in Eq. (1) can be decomposed into N scalar optimization subproblems using Tchebycheff approach in Eq. (2) with the weight vectors w . During the evolutionary search, MOEA/D maintains a population of N points P = {x1, x 2, ... , x N }, where xi ( i ∈ [1, N ]) is aimed at optimizing i-th subproblem. As different subproblems may have different computation difficulties, a dynamic resource assignment for each subproblem is designed in MOEA/D-DRA [22], which computes the utility π i of i-th subproblem as follows.

Q. Lin et al. / Computers & Operations Research 78 (2017) 94–107

⎧ ⎪1 if Δi > 0.001 πi = ⎨ ⎪ i i ⎩ (0.95 + 0.05 × Δ /0.001) × π otherwise

(3)

where Δi is the relative decrease of the objective value in subproblem i, which is defined by

Δi =

(

)

(

)

g tch x old w i, z* − g tch xnew w i, z* g

tch

(x

old

i

)

w , z*

(4)

As the utility π i is updated periodically, x old is the old best solution for i-th subproblem found in the last period, while xnew is the new best solution for i-th subproblem generated in the current period. When the value of Δi is smaller than 0.001, it indicates that the evolutionary search is stagnated in this period. Therefore, the value of π i is reduced in order to save the computational resource.

3. The proposed AMOEA/D algorithm 3.1. The selected DE mutation strategies In AMOEA/D, four classical DE mutation strategies, i.e., DE/rand/ 1, DE/rand/2, DE/current-to-rand/2 and DE/current-to-rand/1, are adopted to compose the candidate operator pool as they have strong exploration capability and distinct search characteristics to prevent premature [30–33]. They are respectively formulated in Eqs. (5)–(8).

vi = x r1 + F × (x r2 − x r3)

(5)

and-error search for the most suitable DE operator and its associated parameter values [47,48]. In this paper, an adaptive operator selection for DE is designed for MOEA/D to tackle MOPs, which is extended from [33]. Four DE strategies, such as DE/rand/1, DE/ rand/2, DE/current-to-rand/1 and DE/current-to-rand/2 are adaptively executed. During the evolutionary procedure, one DE strategy will be selected according to the probability that is obtained by its historically successful experience to produce superior offspring. A larger probability is assigned to the DE mutation strategy that behaves more successfully in the previous generation, while a smaller one is allocated when it cannot effectively produce superior offspring. Therefore, the most suitable DE mutation strategy will be gradually evolved at different learning stages for the problem under consideration. Assume that the probability of applying the k-th DE mutation strategy at generation G is represented by Pk, G (k = 1, 2, 3, 4). At the beginning of evolutionary search, all candidate DE mutation strategies have the equal probabilities to be chosen and their probabilities are all initialized to 0.25. At each generation G , after evaluating the objectives of child solutions, their fitness values are further computed based on the aggregation function as defined in Eq. (2). Then, the number of child solutions, which are produced by the k-th strategy and have better fitness values than their parents, is recorded by successk, G ; the number of other child solutions generated by the k-th strategy is recorded by failk, G . Only these successful solutions will survive in the next generation. After the pre-defined learning period (LP) is achieved, the probabilities of different DE mutation strategies start to be dynamically updated at each subsequent generation, as follows.

Pk, G = vi = x r1 + F × (x r2 − x r3) + F × (x r4 − x r5)

(6)

vi = xi + K × (x r1 − xi ) + F × (x r2 − x r 3) + F × (x r 4 − x r5)

(7)

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p

k, G 4 ∑i = 1 pi, G

(10)

where pk, G is obtained by G−1

vi = xi + F × (x r1 − xi ) + F × (x r2 − x r3)

(8)

where x r1, x r2, x r3, x r4 and x r5 are five distinct random individuals selected from the current population, which are also different from the target vector x i ( i = 1, 2, ... , N ). The parameters K and F are called the scaling factor controlling the mutation scale, which is generally restricted in the range (0, 1]. After the execution of DE mutation, a binomial crossover operator is generally employed on the target vector x i to generate a trial vector x i′. This is achieved by setting the variables of x i′ with the corresponding variables in mutant vector vi or target vector x i , as described in Eq. (9).

⎧ if (ri, j < CRorj = jrand ) ⎪ vi, j xi′, j = ⎨ ⎪ ⎩ xi, j otherwise

(9)

where ri, j ( i ∈ [1, N ] and j ∈ [1, n]) is a uniformly random number in [0, 1], CR ∈ [0, 1] is the crossover rate that controls the proportion of variables inherited from the mutant vector, jrand is a uniformly distributed random integer in [1, n] that makes sure at least one dimension of trial vector is inherited from the mutant vector. 3.2. Adaptive operator selection of DE strategies As mentioned in Sections 2.2 and 2.3, different DE mutation strategies have distinct performance in solving various kinds of optimization problems and thus adaptive control strategy is preferred for DE to substitute the computationally expensive trial-

pk, G =

∑ g = G − LP successk, G G−1

G−1

∑ g = G − LP successk, G + ∑ g = G − LP failk, G

+α (11)

where G > LP and α ¼0.001 is a small constant value to avoid the possible null success rate. Obviously, the larger success rate for the k-th strategy in Eq. (11) during the past LP generations will have the bigger probability to be chosen. Thus, the more suitable DE mutation strategy will be assigned with a larger probability to produce superior solutions. 3.3. Adaptive control of parameter settings In conventional DE strategy, the choice of numerical values for the control parameters F and CR is highly dependent on the target problem. In most variants of MOEA/D [49–51], the settings of F and CR are fixed through all the evolutionary stage. However, this fixed parameter setting may not provide the optimal performance as different values of CR and F in various DE strategies may greatly impact their performance when dealing with various optimization problems. Hence, it is better to dynamically select the values of CR and F for each DE strategy according to the evolutionary situation, and such research studies have been conducted in solving singleobjective optimization problems [48,52]. Here, the adaptive parameter control is extended for tackling MOPs. For each individual xi ( i ∈ [1, N ]), it is associated with its own parameter settings of CRi and Fi to produce the child solution. At the discovery period (the first LP iterations), their parameter settings are randomly initialized in a predefined suitable range. After that, these values of CRi and Fi are updated probabilistically according to the adaptive control parameters μF , i and μCR, i , depending on the following equations.

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Q. Lin et al. / Computers & Operations Research 78 (2017) 94–107

Fig. 1. The pseudo-code of AMOEA/D.

CRi = Gaussian(μCR, i , 0.1)

(12)

Fi = Cauchy(μF , i , 0.1)

(13)

where Gaussian(μ, δ ) and Cauchy(μ, δ ) return the random numbers that are generated from the Gaussian and Cauchy distributions with the mean value μ and variance value δ , respectively. Once the values of CRi and Fi are got out of the range [0,1], it has to be regenerated again. Without loss of generality, the initial values of μCR, i and μF , i are both set to 0.5. Moreover, during the evolutionary phase, the values of μCR, i and μF , i are dynamically updated as follows.

μCR, i = (1 − λ ) × μCR, i + λ × meanW (SCR )

(14)

meanW (SCR ) =

meanL (SF ) =

S

∑k =CR1 ωk × SCR, k

(16)

∑F ∈ S F2 F

∑F ∈ S F

(17)

F

where SCR returns the size of the set SCR and ωk is obtained by

ωk =

Δgktch SCR ∑k = 1 Δgktch

(18)

(

where Δgktch = g tch childk, G − g tch parentk, G

(

)

)

is the increment of

the fitness between the child and the parent solutions. 3.4. The complete AMOEA/D algorithm

μF , i = (1 − λ ) × μF , i + λ × meanL (SF )

(15)

where the sets SCR and SF respectively contain the values of CRi and Fi that are successful to produce the better child solutions in the previous generations; the control parameter λ ¼0.1 is a learning rate; meanW (⋅) is an weighted mean and meanL (⋅) is a Lehmer mean, which are got by

The above subsections have introduced the four selected DE strategies, and the adaptive control strategies for operator selection and parameter settings, which are the main contributions of this paper. Besides that, dynamic resource assignment [22] and stable matching model [6] are also adopted in AMOEA/D to further enhance the comprehensive performance. The pseudo-code of

Q. Lin et al. / Computers & Operations Research 78 (2017) 94–107

AMOEA/D is presented in Fig. 1. In the initialization procedure of line 1 in Fig. 1, N weight vectors are first defined and a population P with N solutions is randomly produced. After that, each weight vector i ( i ∈ [1, N ]) finds its T neighbors based on their Euclidean distances and the set B(i ) includes all the T neighbors of weight vector i. The ideal vector z* is obtained by z*j = min {f j (x ) x ∈ P}, while the nadir vector z nad is got by z nad = max {f j (x ) x ∈ P} for each j ∈ [1, m]. Besides that, j the generation time G, the set A, and π i (i ∈ [1, N ]) are all initialized in line 2. After that, AMOEA/D enters into the loop of evolutionary process until the generation time G reaches the predefined maximum time max_gen, which is expressed in lines 4-33. During the evolutionary process, the dynamic resource assignment is first operated to pick the subproblems for evolution in line 4, where the selected subproblems are preserved in set I. Then, for each subproblem i in I, a random number rand is generated to determine the parent set E that is the entire population or the T neighbors of subproblem i in lines 8–12. After that, the DE operator and its corresponding parameter settings (CR and F) are adaptively selected in lines 13–14, and resultantly a new solution y can be obtained by using the selected DE strategy and polynomial mutation [53] in lines 15–16. If the dimension of y, i.e., yi (i ∈ [1, n]) is out of boundary, it will be reset randomly within the boundary range in line 17. After the objectives of the new solution y are evaluated, the values of z* and z nad are updated in lines 19–26. Then, the offspring population O is generated and a union population U is built by combining the original population P and O in line 28. After that, the stable matching approach [6] is executed on U to select a new population P for the next evolution in line 29. The utility π i of each subproblem i (i ∈ [1, N ]) is updated for each period of 30 generations in lines 31–33, as suggested in MOEA/DSTM [6]. The above evolutionary phase will repeat until the generation times reach max_gen. At the end of algorithm, all the solutions in population P are reported as the final approximated PF.

4. Experimental results In this section, the experimental results of our algorithm are provided and further compared with various MOEA/D variants, e.g., MOEA/D [8], MOEA/D-DE [14], MOEA/D-DE þPSO [19], ENSMOEA/D [16], MOEA/D-FRRMAB [5], and MOEA/D-STM [6]. Firstly, the related experimental settings, such as the standard benchmark problems, performance metric and parameter settings of all the compared algorithms are introduced. Secondly, the experimental results of our algorithm are compared with that of MOEA/D, MOEA/D-DE, MOEA/D-DE þPSO, ENS-MOEA/D, MOEA/D-FRRMAB and MOEA/D-STM. Moreover, the effectiveness of the proposed adaptive control strategies on operator selection and parameter settings is experimentally validated. At last, the best final nondominated sets for all the test problems are plotted, which are very close to the true PFs and distributed uniformly along the true PFs. 4.1. Experimental settings

(1) Standard benchmark problems: in this study, the unconstrained UF test problems (UF1–UF10) [21] are adopted to assess the performance of our algorithm. They are characterized with various complicated PS shapes, which are widely used to test the comprehensive performance of various MOEA/ D variants [5,7,54,55]. It is noted that the dimension of decision variables is set to 30 for all the UF test problems. (2) Performance metric: the main purpose of MOPs includes two

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aspects, i.e., convergence and diversity. That is to say, the obtained approximation set is preferred to approximate the true PF as close as possible and cover the true PF as widely as possible. Therefore, in this paper, inverted generational distance (IGD) [14] is adopted here for this purpose. Although other performance metric, like Hypervolume (HV) [56], can also be used to measure the convergence and diversity, it is pointed out in [57] that the IGD and HV results show high consistencies on convex PFs and certain contradictions on concave PFs. Thus, only the IGD metric is adopted here as a performance metric due to the page limit, and it is still sufficient and convincible for performance comparison as most of our compared algorithms (e.g., MOEA/D, MOEA/DDE, and ENS-MOEA/D) also only adopt the IGD metric in their performance comparison. Assume that P * is a set of optimal solutions uniformly distributed along PF, while S represents the approximated set obtained by multiobjective algorithms. The IGD results can be obtained by

IGD(S, P *) =

∑x ∈ P * dist (x, S ) |P *|

(19)

where dist (x, S ) returns the nearest Euclidean distance from solution x to the points in S and |P *| indicates the size of P *. It is noted that the true PF of each MOP in our experiments is assumed to be known in advance. When the size of P * is large enough to cover the entire true PF, it can effectively measure the convergence and diversity of S. Generally speaking, a lower value of the IGD metric is preferred as it indicates that the obtained set S is closer to the true PF and more uniformly distributed along the true PF. In our experimental studies, the IGD results are obtained by using 1000 sample points that are uniformly distributed along the true PF for the bi-objective test instances, and 10,000 sample points for the three-objective ones. (3) Parameter settings: all the parameters of MOEA/D, MOEA/DDE, MOEA/D-DE þ PSO, MOEA/D-STM, MOEA/D-FRRMAB and ENS-MOEA/D are set as recommended in their corresponding references [5,6,8,14,16,19]. The source codes of MOEA/D, MOEA/D-DE, MOEA/D-STM and MOEA/D-FRRMAB are provided by their authors as implemented in jMetal [58], MOEA/D-DE þPSO is also implemented by us in jMetal, while ENS-MOEA/D is realized by the authors in MATLAB. Besides that, the source code of AMOEA/D is also implemented by us in jMetal and the parameter settings of AMOEA/D are described as follows. The learning period (LP) is set to 50 as suggested in ENS-MOEA/D, which is also verified by our experiments to give a better overall performance. For polynomial mutation, the mutation probability pm is set to 1/n (n is the number of decision variables) and the mutation index η is 20. Regarding to the selected DE strategies in Eqs. (5)–(7), the values of F are randomly generated in (0.5, 0.9) while the values of CR are all initialized in (0.4, 0.6) during the first LP iterations. After that, they are adaptively adjusted according to Eqs. (12)–(18). K is predefined as a fixed value 0.5 in the whole evolution and all the parameters of F, CR and K are all fixedly set to 0.5 for Eq. (8). The neighborhood size T is set to 20, and the probability δ that determines the parents selected from the neighbors or the entire population is set to 0.9. Moreover, the population sizes N for all the algorithms are respectively set to 600 and 1000 for the bi-objective and three-objective test problems. The maximum number of function evaluation is 300,000 and thus the maximum generations for bi-objective and three-objective test problems are respectively 500 and 300. All the compared algorithms are run by 30 independent times

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Table 1 Comparison results of AMOEA/D with other MOEA/D variants. Algorithms Problems UF1

UF2

UF3

UF4

UF5

UF6

UF7

UF8

UF9

UF10

Mean Std Rank Mean Std Rank Mean Std Rank Mean Std Rank Mean Std Rank Mean Std Rank Mean Std Rank Mean Std Rank Mean Std Rank Mean Std Rank

Rank Sum Final Rank

MOEA/D

MOEA/D-DE

MOEA/D þPSO

ENS

STM

FRRMAB

AMOEA

7.25E  02 1.21E  02 7– 3.13E  02 1.35E 02 7 1.52E  01 3.51E  02 7 3.78E  02 1.50E  03 2 2.60E  01 6.99E  02 3 2.17E  01 1.27E 01 6 1.83E  02 1.42E  01 7 7.94E  02 6.20E  03 7 4.46E  02 3.75E  02 7 1.96E  01 4.39E  02 2þ 55 7

1.39E  03 6.83E  05 4 6.77E  03 2.17E  03 5 1.48E  02 1.50E  02 6 6.80E  02 2.84E  03 7 3.61E  01 5.81E  02 6 6.46E  02 1.17E  01 4 1.50E  03 3.01E  04 4 4.27E 02 8.02E  03 5 2.59E  02 2.86E  02 5 3.97E 01 4.82E  02 7 53 5

1.83E  02 2.36E  02 6 1.17E  02 7.56E  04 6 5.83E  03 1.83E  03 4 5.46E  02 4.05E  03 5 4.54E  01 1.19E  01 7 8.23E  01 1.57E  01 7 8.78E  03 2.37E  01 6 7.62E  02 6.53E  03 6 3.90E  02 3.50E  02 6 1.87E  01 1.55E  03 1þ 54 6

1.94E  03 3.17E  04 5 4.80E  03 2.07E 03 4 6.27E  03 3.42E  03 5 6.11E  02 3.93E  03 6 2.61E  01 9.82E  02 4 1.70E 01 2.63E  01 5 1.75E  03 3.72E  04 5 3.28E  02 7.65E  03 4 2.41E  02 5.64E  02 4 3.14E  01 7.48E  02 4 46 4

1.11E  03 7.46E  05 3 2.87E  03 1.56E  03 3 4.63E  03 5.92E  03 2 5.13E  02 3.27E 03 3 2.42E  01 2.43E  02 2 6.45E  02 1.87E  02 3 1.13E  03 7.20E  05 2þ 1.89E  02 7.77E 04 2E 1.69E  02 5.63E  04 2 3.88E  01 6.05E  02 6 28 2

1.02E  03 1.11E  04 2E 1.70E 03 5.65E  04 1E 5.10E  03 4.86E  03 3 5.31E  02 3.82E  03 4 2.97E  01 5.31E  02 5 6.38E  02 4.37E  02 2 1.09E  03 1.64E  04 1þ 3.22E  02 2.68E  03 3 2.16E  02 4.66E  02 3– 3.84E  01 7.57E  02 5 29 3

1.01E  03 4.73E  05 1 2.03E  03 7.41E  04 2 1.54E  03 6.22E  03 1 3.47E  02 1.78E  03 1 1.61E  01 5.42E  02 1 5.48E  02 4.79E  02 1 1.18E  03 9.09E  05 3 1.60E  02 3.41E  03 1 1.59E  02 3.67E 04 1 2.28E  01 8.64E  02 3 15 1

Table 2 Average rankings of all the MOEA/D variants by the Friedman test for all the test problems. Algorithms Problems UF1 UF2 UF3 UF4 UF5 UF6 UF7 UF8 UF9 UF10 Average Ranking

Ranking Ranking Ranking Ranking Ranking Ranking Ranking Ranking Ranking Ranking

MOEA/D

MOEA/D-DE

MOEA/DE þ PSO

ENS

STM

FRRMAB

AMOEA

7.000 6.967 7.000 1.933 2.767 5.200 7.333 6.600 5.467 1.933 4.4665

1.867 4.333 4.267 5.767 4.800 3.200 4.033 4.767 4.200 5.867 3.867

6.000 6.033 3.667 2.967 6.933 6.933 6.667 6.400 5.100 1.800 3.900

4.000 3.900 2.833 3.900 2.867 3.700 4.000 3.567 3.733 4.300 4.150

2.600 2.767 2.633 2.400 2.433 2.333 2.033 1.567 1.900 5.567 4.084

1.633 1.500 2.500 2.700 3.400 2.367 1.667 3.433 3.167 5.800 3.717

1.767 1.833 2.033 1.000 1.300 1.600 2.300 1.433 1.200 2.733 2.250

on each test problem. 4.2. Performance comparisons with other recent MOEA/D variants In this subsection, the performance of AMOEA/D is compared with the other six MOEA/D variants, i.e., MOEA/D, MOEA/D-DE, MOEA/D-DE þ PSO, ENS-MOEA/D, MOEA/D-FRRMAB and MOEA/DSTM on the UF test problems. It is noted that the above seven algorithms are shortly represented by AMOEA, MOEA/D, MOEA/DDE, MOEA/D þPSO, ENS, STM and FRRMAB in Table 1, which illustrates the mean IGD values obtained from 30 independent runs. The best result on each test problem is highlighted with boldface. Moreover, to identify the significance differences of the results obtained by AMOEA/D and other algorithms on each problem, the Wilcoxon's rank sum test is carried out as shown in Table 1, where

the “þ”, “  ”, and “E ” indicate the results obtained by the algorithm are significantly better than, worse than, and similar to the ones obtained by AMOEA/D using the Wilcoxon's rank sum test with a significant level α ¼0.05. It is clearly demonstrated that AMOEA/D performs best on most of test problems, such as UF1, UF3-UF6, UF8 and UF9, FRRMAB is the best on UF2 and UF7, while MOEA/D-PSO obtains the best result on UF10. The Wilcoxon's rank sum test indicates that the performance of AMOEA/D is statistically similar to MOEA/D-STM on UF8, and to MOEA/D-FRRMAB on UF1 and UF2. These comparison results for the compared algorithms on all the UF test problems are summarized in the last two rows of Table 1, in which the “Rank Sum” row summarizes all the ranks on all the test problems and the “Final rank” row gives the performance ranking according to the “Rank Sum” row. In this way, it is straightforward to find out that AMOEA/D performs best when

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AMOEA/D MOEA/D-STM MOEA/D-FRRMAB ENS-MOEA/D MOEA/D MOEA/D-DE MOEA/D-DE+PSO

0 10 20 30 40 50 Number of Function Evaluation(x6000) Fig. 2. Evolutionary curves of the IGD metric values versus the number of function evaluation.

considering all the UF problems, as it obtains the first rank. For the rest algorithms, MOEA/D-STM and MOEA/D-FRRMAB obtain the second and third ranks, respectively, while ENS-MOEA/D, MOEA/ D-DE, MOEA/D-DE þ PSO and MOEA/D respectively obtain the 4th, 5th, 6th and 7th ranks. Moreover, the Friedman test is further used as a multi-methods & multi-problems approach, to give the performance ranking of each algorithm on all the test problems. The corresponding results obtained by the Friedman test are listed in Table 2, where the best result for each test problem is also highlighted in boldface. Thus, it is easy to find out that AMOEA/D also performs better on more than half of the test problems (e.g., UF3-UF6 and UF8-UF9), which

confirms the superior performance of AMOEA/D over other MOEA/ D variants. In the last row of Table 2, the average ranking of each algorithm on all the test problems is summarized. As indicated by this average ranking, AMOEA/D also obtains the best performance ranking among all the compared algorithms when considering all the test problems. To further investigate the convergence behavior of all the compared algorithms, Fig. 2 illustrates the plots of average IGD results obtained by all the algorithms versus the number of function evaluations on each test problem. In order to distinguish the final experimental results on certain test problems, some subplots are further provided in Fig. 2. Obviously, based on the observation of Fig. 2, it is shown that the IGD results of

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Fig. 3. Box plots of the IGD results obtained by op1, opt2, opt3, op4 and AMOEA/D.

4.3. The effectiveness of adaptive operator selection

Table 3 Comparison summary of AMOEA/D with opt1, opt2, opt3 and opt4.

IGD

 þ E

opt1

opt2

opt3

opt4

8 1 1

9 0 1

8 2 0

8 1 1

AMOEA/D are gradually reduced during the entire evolutionary search, which indicates that our algorithm can effectively avoid stagnation and premature convergence. Moreover, the convergence speed of AMOEA/D is also justified to be faster than that of MOEA/D, MOEA/D-DE, MOEA/D-DE þ PSO, ENS-MOEA/D, MOEA/ D-FRRMAB and MOEA/D-STM on most of test problems.

In our algorithm, four DE strategies are used to compose a DE operator pool, in which the designed adaptive operator selection chooses the suitable DE strategy. To validate the effectiveness of adaptive operator selection in AMOEA/D, AMOEA/D is further compared with its four variants, i.e., opt1, opt2, opt3 and opt4, which share the similar procedures with AMOEA/D except for the adaptive selection strategy on DE operators. It is noted that opt1, opt2, opt3 and opt4 respectively adopt DE/rand/1, DE/current-torand/1, DE/current-to-rand/2 and DE/rand/2 as their evolutionary operators, which also use the same adaptive control strategy on parameter settings. Fig. 3 gives the experimental comparison using the box plots, where their IGD results for 30 independent runs are illustrated. According to the results in Fig. 3, it is illustrated that the performance of AMOEA/D is better on most of test problems,

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Fig. 4. The usage probability of each strategy during the evolutionary process.

such as UF1, UF2, UF4, UF5, UF9 and UF10. This is because AMOEA/ D always chooses the most appropriate DE operator for the current generation. Moreover, the Wilcoxon's rank sum test indicates that AMOEA/D performs similarly to opt1 and opt2 on UF8, and to opt4 on UF6. Besides that, AMOEA/D is only worse than opt1 on UF7, than opt3 on UF3 and UF6, than opt4 on UF3. For the remaining comparisons, AMOEA/D performs better. The corresponding comparison summary is provided in Table 3, where ‘  ’, ‘ þ ’ and ‘ E’ respectively indicate the number of test problems that the

results obtained by the corresponding algorithm are worse than, better than or similar with that of AMOEA/D according to the Wilcoxon's rank sum test at a 0.05 significance level. From Table 3, the effectiveness of adaptive operator selection in our algorithm is validated, as AMOEA/D respectively performs better than or similarly with opt1, opt2, opt3 and opt4 on 9, 10, 8, 9 out of 10 UF test problems. In order to further have a deeper recognition of the behavior of AMOEA/D, we further plot the usage probability of each DE

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exp3 AMOEA

exp2

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exp2

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exp1

exp2

exp3 AMOEA

UF10

2.5 2 1.5 1 0.5 exp1

exp2

exp3 AMOEA

Fig 5. Box plots of the IGD results obtained by exp1, exp2, exp3 and AMOEA/D.

Table 4 Comparison summary of AMOEA/D with exp1, exp2 and exp3.

IGD

 þ E

exp1

exp2

exp3

6 1 3

2 0 8

5 1 4

strategy during the evolutionary process in Fig. 4. The whole 300,000 function evaluations are divided into 50 consecutive search phases, and each of them includes 6000 function evaluations. By this way, the usage probability of each strategy during each phase can be calculated. In Fig. 4, opt1, opt2, opt3 and opt4 respectively indicate the usage of DE/rand/1, DE/current-to-rand/1, DE/current-to-rand/2 and DE/rand/2 in AMOEA/D. Based on the

observation from Fig. 4, it is shown that no single DE strategy can be consistently used over the whole search process on all the test problems, which in turn proves the self-adaptive ability of our algorithm to adjust the used DE strategy. It is noted that some DE strategies may be preferred at the beginning of evolutionary process. For example, on UF1, the opt2 strategy is frequently used during 1–4 search phases. Regarding other test problems, similar phenomenon can be found. Anyway, these plots in Fig. 4 validate the effectiveness of AMOEA/D in adaptively switching the usage of different DE strategies in different search phases. 4.4. The validity of adaptive parameter control To further investigate the validity of adaptive control on parameter settings, AMOEA/D is compared to other three AMOEA/D

Q. Lin et al. / Computers & Operations Research 78 (2017) 94–107

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variants, i.e., exp1, exp2 and exp3. It is noted that exp1, exp2 and exp3 follow the similar components with AMOEA/D, except that exp1 adopts the fixed parameter settings for CR and F, exp2 uses the fixed parameter setting for F while exp3 utilizes the fixed parameter setting for CR. Thus, by comparing the performance of AMOEA/D with exp1, exp2 and exp3, it can effectively study the impact of adaptive control strategy on parameter settings. Fig. 5 gives the box plots of IGD results obtained by exp1, exp2, exp3 and AMOEA/D, which shows that AMOEA/D can obtain the better results on most of test problems by using the adaptive control on both of F and CR. Moreover, the Wilcoxon's rank sum test states that AMOEA/D gets the similar results with exp1 on UF3, UF6 and UF7, with exp2 on UF1-UF4, UF6, UF8-UF10, and with exp3 on UF2, UF3, UF6 and UF7. These comparison results of AMOEA/D with exp1, exp2 and exp3 on all the UF test problems are summarized in Table 4, where ‘  ’, ‘ þ’ and ‘ E’ respectively represent the number of test problems that the results obtained by the corresponding variant are worse than, better than or similar with that of AMOEA/D according to the Wilcoxon's rank sum test at a 0.05 significance level. From Table 4, AMOEA/D performs better than or similarly with exp1 and exp3 on 9 out of 10 test problems, and better than or similarly with exp2 on all the test problems. More specifically, AMOEA/D is better than exp1 on UF1, UF2, UF4, UF5, UF9 and UF10, than exp2 on UF5 and UF7, than exp3 on UF1, UF4, UF5, UF9 and UF10; while AMOEA/D performs only worse than exp1 and exp3 on UF8. Therefore, the above experimental results justify that the adaptive control strategy on F and CR is effective to enhance the performance of our algorithm. 4.5. The best approximation obtained by our algorithm In the above subsections, the advantages of our algorithm are confirmed and the effectiveness of adaptive control strategies on operator selection and parameter settings is validated. In order to have a visual observation on the performance of our algorithm, Fig. 6 gives the plots of best approximation set obtained by AMOEA/D (shortened as AMOEA) in 30 independent runs, in which the true PFs are also illustrated for comparison. From Fig. 6, it clearly indicates that the final nondominated sets found by AMOEA/D are very close to the true PFs and distributed uniformly along the true PFs, except for UF5, UF6 and UF10. In fact, as UF5 and UF6 are characterized with discontinuous points that are very difficult to be optimized, all the compared algorithms, such as MOEA/D, MOEA/D-DE, MOEA/D-DE þPSO, ENS-MOEA/D, MOEA/DSTM and MOEA/D-FRRMAB, cannot solve them very well; while UF10 owns many local PFs, which requires more computational effort for AMOEA/D to approximate its true PF very well.

5. Conclusions In this paper, a novel adaptive MOEA/D algorithm is designed, which performs the adaptive control strategies on both of operator section and parameter settings. Based on the historically successful experience, the suitable DE mutation operators are adaptively determined and the associated parameter settings are also dynamically adjusted using the former parameter settings that have higher probability to generate the superior offspring. By this way, both of DE operators and their parameter settings are automatically tuned using the feedback information from the evolutionary search. The performance of our algorithm is evaluated by the complicated UF test problems and the experimental results validate that the nondominated solutions found by AMOEA/D are distributed uniformly and close to the true PFs. When compared to various MOEA/D variants, such as MOEA/D, MOEA/D-DE, MOEA/DDE þPSO, ENS-MOEA/D, MOEA/D-STM and MOEA/D-FRRMAB,

AMOEA/D performs better on most of test problems. Besides that, the effectiveness of adaptive control strategies on operator selection and parameter settings is also validated experimentally. In our future work, we will further study the self-adaptability of AMOEA/D. Some remaining fixed parameter settings, such as the neighboring size T and the weight vectors, can be modified to be self-adjusted to further enhance the performance of AMOEA/D. There are also some DE adaptation approaches designed for solving single-objective optimization problems, such as that presented in L-SHADE [59] and MOS [60]. These DE adaptation methods can be further studied by extending them to solve MOPs. At last, AMOEA/D will be also investigated to tackle some practical engineering problems.

Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61402291, Seed Funding from Scientific and Technical Innovation Council of Shenzhen Government under Grant 0000012528, Foundation for Distinguished Young Talents in Higher Education of Guangdong under Grant 2014KQNCX129, and Natural Science Foundation of SZU under Grant 201531.

References [1] Ishibuchi H, Murata T. A multi-objective genetic local search algorithm and its application to flowshop scheduling. IEEE Trans Syst Man Cybern Part C: Appl Rev 1998;28(3):392–403. [2] Qi Y, Hou Z, Li H, Huang J, Li X. A decomposition based memetic algorithm for multi-objective vehicle routing problem with time windows. Comput Oper Res 2015;62:61–77. [3] Garza-Fabre M, Rodriguez-Tello E, Toscano-Pulido G. Constraint-handling through multi-objective optimization: the hydrophobic-polar model for protein structure prediction. Comput Oper Res 2015;53:128–53. [4] Deb K. Multi-objective optimization using evolutionary algorithms.New York, USA: John Wiley & Sons; 2001. [5] Li K, Fialho A, Kwong S, Zhang Q. Adaptive operator selection with bandits for a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evolut Comput 2014;18(1):114–30. [6] Li K, Zhang Q, Kwong S, Li M, Wang R. Stable matching-based selection in evolutionary multiobjective optimization. IEEE Trans Evolut Comput 2014;18 (6):909–23. [7] Ma X, Liu F, Qi Y, Gong M, Yin M, Li L, et al. MOEA/D with opposition-based learning for multiobjective optimization problem. Neurocomputing 2014;146:48–64. [8] Zhang Q, Li H. MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evolut Comput 2007;11(6):712–31. [9] Tan YY, Jiao YC, Li H, Wang XK. MOEA/D þ uniform design: a new version of MOEA/D for optimization problems with many objectives. Comput Oper Res 2013;40(6):1648–60. [10] Ma X, Qi Y, Li L, Liu F, Jiao L, Wu J. MOEA/D with uniform decomposition measurement for many-objective problems. Soft Comput 2014;18(12):2541– 64. [11] Wang R, Zhang T, Guo B. An enhanced MOEA/D using uniform directions and a pre-organization procedure. In: 2013 IEEE congress on evolutionary computation (CEC). 2013. p. 2390–7. [12] Jiang S, Cai Z, Zhang J, Ong YS. Multiobjective optimization by decomposition with pareto-adaptive weight vectors. In: Proceedings of the 2011 seventh international conference on natural computation (ICNC), Vol. 3; 2011. p. 1260– 4. [13] Qi Y, Ma X, Liu F, Jiao L, Sun J, Wu J. MOEA/D with adaptive weight adjustment. Evolut Comput 2014;22(2):231–64. [14] Li H, Zhang Q. Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans Evolut Comput 2009;13(2):284–302. [15] Venske SMS, Gonçalves RA, Delgado MR. ADEMO/D: multiobjective optimization by an adaptive differential evolution algorithm. Neurocomputing 2014;127:65–77. [16] Zhao SZ, Suganthan PN, Zhang Q. Decomposition-based multiobjective evolutionary algorithm with an ensemble of neighborhood sizes. IEEE Trans Evolut Comput 2012;16(3):442–6. [17] Li K, Kwong S, Zhang Q, Deb K. Interrelationship-based selection for decomposition multiobjective optimization. IEEE Trans Cybern. Accepted. doi: 〈http://dx.doi.org/10.1109/TCYB.2014.2365354http://dx.doi.org/10.1109/TCYB. 2014.2365354〉.

Q. Lin et al. / Computers & Operations Research 78 (2017) 94–107

[18] Li K, Deb K, Zhang Q, Kwong S. Combining dominance and decomposition in evolutionary many-objective optimization. IEEE Trans Evolut Comput. Accepted. doi: 〈http://dx.doi.org/10.1109/TCYB.2014.236535410.1109/TEVC.2014. 2373386http://dx.doi.org/10.1109/TCYB.2014.236535410.1109/TEVC.2014. 2373386〉. [19] Mashwani WK, Salhi A. Multiobjective memetic algorithm based on decomposition. Appl Soft Comput 2014;21:221–43. [20] Ke L, Zhang Q, Battiti R. MOEA/D-ACO: a multiobjective evolutionary algorithm using decomposition and AntColony. IEEE Trans Cybern 2013;43 (6):1845–59. [21] Zhang Q, Zhou A, Zhao S, Suganthan PN, Liu W, Tiwari S. Multiobjective optimization test instances for the CEC 2009 special session and competition. University of Essex, Colchester, UK and Nanyang Technological University, Singapore, Special Session on Performance Assessment of Multi-Objective Optimization Algorithms, Technical Report; 2008.p. 1–30. [22] Zhang Q, Liu W, Li H. The performance of a new version of MOEA/D on CEC09 unconstrained MOP test instances. In: IEEE congress on evolutionary computation, Vol. 1; 2009. p. 203–8. [23] Miettinen K. Nonlinear multiobjective optimization.New York, USA: International Series in Operations Research & Management Science, vol. 12, Springer Science & Business Media; 1998. [24] Medina MA, Das S, Coello CAC, Ramírez JM. Decomposition-based modern metaheuristic algorithms for multi-objective optimal power flow–a comparative study. Eng Appl Artif Intell 2014;32:10–20. [25] Segundo GAS, Krohling RA, Cosme RC. A differential evolution approach for solving constrained min-max optimization problems. Expert Syst Appl 2012;39(18):13440–50. [26] Hao JH, Liu M, Lin JH, Wu C. A hybrid differential evolution approach based on surrogate modelling for scheduling bottleneck stages. Comput Oper Res 2016;66:215–24. [27] dos Santos Coelho L, Mariani VC. Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valvepoint effect. IEEE Trans Power Syst 2006;21(2):989–96. [28] Alatas B, Akin E, Karci A. MODENAR: multi-objective differential evolution algorithm for mining numeric association rules. Appl Soft Comput 2008;8 (1):646–56. [29] Tsai JT, Fang JC, Chou JH. Optimized task scheduling and resource allocation on cloud computing environment using improved differential evolution algorithm. Comput Oper Res 2013;40(12):3045–55. [30] Storn R, Price K. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 1997;11(4):341–59. [31] Gämperle R, Müller SD, Koumoutsakos P. A parameter study for differential evolution. Adv Intell Syst Fuzzy Syst Evolut Comput 2002;10:293–8. [32] Iorio AW, Li X. Solving rotated multi-objective optimization problems using differential evolution. In: AI 2004: Advances in artificial intelligence. Springer, Berlin Heidelberg; 2004. p. 861–72. [33] Qin AK, Huang VL, Suganthan PN. Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans Evolut Comput 2009;13(2):398–417. [34] Gong W, Fialho Á, Cai Z. Adaptive strategy selection in differential evolution. In: Proceedings of the 12th annual conference on Genetic and evolutionary computation, ACM; 2010. p. 409–16. [35] Gong W, Fialho Á, Cai Z, Li H. Adaptive strategy selection in differential evolution for numerical optimization: an empirical study. Inf Sci 2011;181 (24):5364–86. [36] Pan QK, Suganthan PN, Wang L, Gao L, Mallipeddi R. A differential evolution algorithm with self-adapting strategy and control parameters. Comput Oper Res 2011;38(1):394–408. [37] Gong W, Cai Z, Ling CX, Li C. Enhanced differential evolution with adaptive strategies for numerical optimization. IEEE Trans Syst Man Cybern Part B: Cybern 2011;41(2):397–413.

107

[38] Zhang J, Sanderson AC. JADE: adaptive differential evolution with optional external archive. IEEE Trans Evolut Comput 2009;13(5):945–58. [39] Brest J, Greiner S, Boskovic B, Mernik M, Zumer V. Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans Evolut Comput 2006;10(6):646–57. [40] Das S, Suganthan PN. Differential evolution: a survey of the state-of-the-art. IEEE Trans Evolut Comput 2011;15(1):4–31. [41] Yu WJ, Shen M, Chen WN, Zhan ZH, Gong YJ, Lin Y, et al. Differential evolution with two-level parameter adaptation. IEEE Trans Cybern 2014;44(7):1080–99. [42] Eiben ÁE, Hinterding R, Michalewicz Z. Parameter control in evolutionary algorithms. IEEE Trans Evolut Comput 1999;3(2):124–41. [43] Wang Y, Cai Z, Zhang Q. Differential evolution with composite trial vector generation strategies and control parameters. IEEE Trans Evolut Comput 2011;15(1):55–66. [44] Mallipeddi R, Suganthan PN, Pan QK, Tasgetiren MF. Differential evolution algorithm with ensemble of parameters and mutation strategies. Appl Soft Comput 2011;11(2):1679–96. [45] Yang Z, Tang K, Yao X. Self-adaptive differential evolution with neighborhood search. In: 2008 IEEE congress on evolutionary computation; 2008. p. 1110–6. [46] Shang Z, Li Z, Liang JJ, Niu B. Control parameters self-adaptation in differential evolution based on intrisic structure information. In: 2012 IEEE congress on evolutionary computation (CEC); 2012. p. 1–6. [47] Kovačević D, Mladenović N, Petrović B, Milošević P. DE-VNS: self-adaptive differential evolution with crossover neighborhood search for continuous global optimization. Comput Oper Res 2014;52:157–69. [48] Mallipeddi R, Wu G, Lee M, Suganthan PN. Gaussian adaptation based parameter adaptation for differential evolution. In: 2014 IEEE congress on evolutionary computation (CEC); 2014. p. 1760–7. [49] Zhang Q, Liu W, Tsang E, Virginas B. Expensive multiobjective optimization by MOEA/D with Gaussian process model. IEEE Trans Evolut Comput 2010;14 (3):456–74. [50] Tan YY, Jiao YC, Li H, Wang XK. A modification to MOEA/D-DE for multiobjective optimization problems with complicated Pareto sets. Inf Sci 2012;213:14–38. [51] Dai C, Wang Y. A new decomposition based evolutionary algorithm with uniform designs for many-objective optimization. Appl Soft Comput 2015;30:238–48. [52] Tanabe R, Fukunaga A. Success-history based parameter adaptation for differential evolution. In: 2013 IEEE congress on evolutionary computation (CEC); 2013. p. 71–8. [53] Deb K, Goyal M. A combined genetic adaptive search (geneas) for engineering design. Comput Sci Inform 1996;26:30–45. [54] Khan W, Zhang Q. MOEA/D-DRA with two crossover operators. In: 2010 UK workshop on computational intelligence (UKCI); 2010. p. 1–6. [55] Tan YY, Jiao YC, Li H, Wang XK. MOEA/D-SQA: a multi-objective memetic algorithm based on decomposition. Eng Optim 2012;44(9):1095–115. [56] Zitzler E, Thiele L, Laumanns M, Fonseca CM, Da Fonseca VG. Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evolut Comput 2003;7(2):117–32. [57] Jiang SW, Ong Y-S, Zhang J, Feng L. Consistencies and contradictions of performance metrics in multiobjective optimization. IEEE Trans Cybern 2014;44 (12):2391–404. [58] Durillo JJ, Nebro AJ. jMetal: a Java framework for multi-objective optimization. Adv Eng Softw 2011;42(10):760–71. [59] Tanabe R, Fukunaga AS. Improving the search performance of SHADE using linear population size reduction. In: 2014 IEEE congress on evolutionary computation (CEC); 2014. p. 1658–65. [60] LaTorre A, Muelas S, Peña JM. Large scale global optimization: Experimental results with MOS-based hybrid algorithms. In: 2013 IEEE congress on evolutionary computation (CEC); 2013. p. 2742–9.