A cloud model based fruit fly optimization algorithm

Knowledge-Based Systems xxx (2015) xxx–xxx

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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

A cloud model based fruit fly optimization algorithm q Lianghong Wu ⇑, Cili Zuo, Hongqiang Zhang School of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan, Hunan 411201, PR China

a r t i c l e

i n f o

Article history: Received 6 February 2015 Received in revised form 1 September 2015 Accepted 7 September 2015 Available online xxxx Keywords: Fruit fly optimization algorithm Cloud model Randomness Fuzziness

a b s t r a c t Fruit Fly Optimization Algorithm (FOA) is a new global optimization algorithm inspired by the foraging behavior of fruit fly swarm. However, similar to other swarm intelligence based algorithms, FOA also has its own disadvantages. To improve the convergence performance of FOA, a normal cloud model based FOA (CMFOA) is proposed in this paper. The randomness and fuzziness of the foraging behavior of fruit fly swarm in osphresis phase is described by the normal cloud model. Moreover, an adaptive parameter strategy for Entropy En in normal cloud model is adopted to improve the global search ability in the early stage and to improve the accuracy of solution in the last stage. 33 benchmark functions are used to test the effectiveness of the proposed method. Numerical results show that the proposed CMFOA can obtain better or competitive performance for most test functions compared with three improved FOAs in recent literatures and seven state-of-the-arts of intelligent optimization algorithm. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Swarm intelligence is a recent trend in computational intelligence and popular for the simplicity of its realizations and effectiveness of solving complex optimization problems. It is the collective behavior of decentralized and self-organized systems, natural or artificial. Various swarm intelligence based algorithms have been proposed in recent years such as particle swarm optimization (PSO) [1], artificial bee colony optimization (ABC) [2], ant colony optimization (ACO) [3], artificial fish swarm algorithm (AFSA) [4], artificial immune systems (AIS) [5], gray wolf optimizer (GWO) [6], multi-verse optimizer (MVO) [7], and ant lion optimizer (ALO) [8], etc. As stochastic optimization techniques, swarm intelligence based methods have been widely applied to many aspects in real-world application and have obtained very promising results. Fruit Fly Optimization Algorithm (FOA), proposed by Pan in 2011 [9], is a new global optimization algorithm inspired by the foraging behavior of fruit fly. Compared with other swarm intelligence based algorithms, FOA has the advantages of being easy to understand and a simple computational process [11,12]. As a novel optimization algorithm, FOA has gained much attention and suc-

q This work is partially supported by National Natural Science Foundation of China (Nos. 61203309, 51374107, 61403134), the Scientific Research Fund of Hunan Provincial Education Department (No. 12B043) and Hunan Provincial Innovation Foundation For Postgraduate (CX2015B488). ⇑ Corresponding author. E-mail addresses: [email protected] (L. Wu), [email protected] (C. Zuo).

cessfully applied in many areas in recent years, such as the financial distress model solving [10], the annual power load forecasting [11], analysis of the service satisfaction in web auction logistics service [12], tuning of PID controller [13,14] and fractional fuzzyPID controller [15], neural network [16], the multidimensional knapsack problem [17], parameter identification of synchronous generator [18], and joint replenishment problems [19]. However, similar to other swarm intelligence based algorithms, FOA also has its own disadvantages. According to an empirically study in [20], the original FOA cannot solve multi-modal optimization problems effectively. To enhance the optimization performance of FOA, some researchers proposed various improved methods. Shan et al. [20] studied that FOA includes a nonlinear generation mechanism of candidate solution which limits the performance of FOA. In order to enhance the performance of FOA, the nonlinear generation mechanism of candidate solution is replaced with a linear generation mechanism of candidate solution, and then a LGMS-based improved FOA is proposed in [20]. In regards to the problem of smell concentration judgment value S is non-negative and that will restrict the application of FOA in some problem, Dai et al. [21] proposed an improved FOA by judging the location of the fruit fly in the quadrant of coordinate system. To overcome the deficiencies of non-negative fitness function, Pan [22] also proposed a modified fruit fly optimization algorithm (MFOA) which included an escape parameter that enabled it to escape from the local extreme solution to find out the global extreme solution. Moreover, since the original FOA searches for global optimal in two dimensional space, it could possibly lead to difficulties in searching for the optimal values in three-dimensional space. Hence,

http://dx.doi.org/10.1016/j.knosys.2015.09.006 0950-7051/Ó 2015 Elsevier B.V. All rights reserved.

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L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx

in the MFOA the original FOA was corrected and extended to the three-dimensional space. Recently, to avoid being trapped in local optimal or premature in multi-modal optimization problems, Yuan et al. [18] proposed a modified fruit fly optimization algorithm (MSFOA) with techniques of multi-swarm strategy, revised evaluation function and shrinking of exploring radius. To eliminate the drawbacks lies with fixed values of search radius of FOA, Pan et al. [23] also introduced an improved fruit fly optimization (IFFO) with a new control parameter and an effective solution generating method to improve the effectiveness of the fruit fly optimization and used it to solve high-dimensional continuous function optimization problems. One of the common modifications in MSFOA [18] and IFFO [23] is that both the distance Disti and smell concentration judgment value Si are removed, and the fitness function is evaluated by the decision variables directly. This modification is helpful for improving the original FOA’s global convergence ability. Recently, Wang et al. [19] also proposed an improved FOA (IFOA) by introducing a method of maintaining the population diversity to enhance the exploration ability and a new parameter to avoid the acquisition of local optimal solution. Experimental results of 18 Benchmark functions and application of joint replenishment problems show that the IFOA has better comprehensive performance than the original FOA and several other competitors. In this paper, an improved fruit fly optimization algorithm based on cloud model, namely CMFOA, is developed. In CMFOA, the normal cloud generator [24] is used to generate a new location for the fruit flies in the osphresis foraging phase. The motivation is that the foraging behavior of fruit fly swarm has features of randomness and fuzziness in the process of fruit flies learning to the best individual to fly a new promising food source. The randomness and fuzziness can be simultaneously described by the normal cloud model [25]. So, to improve the convergence performance of the FOA algorithm the normal cloud generator is used to generate new fruit fly swarms. In addition, a parameter adaptive strategy is introduced to dynamically tune the search range around the swarm location according to the evolution process. To evaluate the effectiveness of the proposed method, the CMFOA is applied to 33 benchmark functions and compared with the basic FOA and several recent variants of FOA such as MSFOA [18], MFOA [22] and IFFO [23]. Numerical results indicate that the CMFOA is able to greatly enhance the convergence performance of the original FOA. To further validate the results, the CMFOA is compared with seven representative swarm intelligence based algorithms and evolutionary algorithms such as GWO[6], MVO[7], ALO[8], CLPSO[28], IASFA[29], SGHS[30] and SaDE[31]. The results prove that the proposed algorithm is able to provide very competitive results compared to these meta-heuristics. The rest of this paper is organized as follows. Section 2 briefly introduces the original FOA and the related works. In Section 3 the normal cloud generator is presented, then, the proposed CMFOA is presented in detail. Experimental design and numerical comparisons are illustrated in Section 4. Finally, Section 5 gives the concluding remarks.

to other species in vision and osphresis. When a fruit fly decides to go for hunting, it will fly randomly to find the location guided by a particular odor. While searching, a fruit fly also sends and receives information from its neighbors and makes comparison with the so far best location and fitness [18]. The food finding process of fruit fly is as follows: firstly, it smells the food source by using osphresis, and flies towards that location; secondly, after it gets close to the food location, the sensitive vision is also used for finding food and other fruit flies’ flocking location, and then it flies towards that direction. According to the food finding procedure of fruit fly swarm, the FOA can be divided into three parts: parameters and population location initialization, osphresis search phase and vision search phase. The Pseudo-code of FOA is showed in Fig. 1. 2.1.1. Parameters and population location initialization The main parameters of the FOA are the maximum iteration number T, the population size NP, and the random flight distance range randValue. The fruit fly swarm location (X axis; Y axis) is randomly initialized in the search space as follows.

X axis ¼ rand  ðUB  LBÞ þ LB

ð1Þ

Y axis ¼ rand  ðUB  LBÞ þ LB

ð2Þ

where rand is a random function which returns a value from the uniform distribution on the interval [0, 1], the UB and LB are the upper and lower bounds of fruit fly swarm location in twodimensional searching space, respectively.

2. Related works In this section, the original FOA is presented firstly. Then, several recent improved FOAs that obtained impressive results are briefly discussed. 2.1. Fruit fly optimization algorithm The FOA is a new method for searching global optimization based on the food foraging behavior of the fruit fly. Fruit flies live in the temperate and tropical climate zones, and they are superior

Fig. 1. Pseudo-code of FOA.

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2.1.2. Osphresis search In the osphresis search phase, a population of NP food sources is generated randomly around the current fruit fly swarm location. Firstly, the random flight direction and distance of an individual fruit fly for food finding are given as follows.

X i ¼ X axis þ randValue

ð3Þ

Y i ¼ Y axis þ randValue

ð4Þ

where i ¼ 1; 2; . . . ; NP, the randValue is a random value in the range of [1, 1]. Then, the distance of the food location to the origin (Disti ) is calculated for each individual, and the smell concentration judgment value (Si ) is further calculated. The value of Si is the reciprocal of the distance of food location to the origin.

Disti ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðX 2i þ Y 2i Þ

ð5Þ

Si ¼ 1=Disti

ð6Þ

Finally, the smell concentration (Smelli ) of the individual fruit fly location by substituting smell concentration judgment value (Si ) into the smell concentration judgment function (or called fitness function) is calculated.

Smelli ¼ FitnessðSi Þ

ð7Þ

2.1.3. Vision search In the vision searching phase, FOA carries out a greedy selection procedure. Firstly, the individual with the maximum smell concentration (the maximum value of Smell) is found out among the fruit fly swarm.

½bestSmell; bestIndex ¼ min ðSmellÞ

ð8Þ

Then, the current maximum smell concentration value (bestSmell) is compared with the value in history (smellBest). If bestSmell < smellBest, smellBest is updated with bestSmell and the fruit fly swarm flies towards that location with the maximum smell concentration value by using vision.

smellBest ¼ bestSmell

ð9Þ

X axis ¼ XðbestIndexÞ

ð10Þ

Y axis ¼ YðbestIndexÞ

ð11Þ

The osphresis searching phase and vision searching phase are repeated, until the smell concentration is not superior to the previous iterative smell concentration any more, or the iterative number reaches the maximum iterative number. 2.2. Modified fruit fly optimization algorithm (MFOA) Just as stated in [22], FOA is likely to get into the local extreme due to the FOA fitness function. According to the original definition of fitness function, FOA did not have high probability of mutation. Consequently, the searching space would be limited, and it would be difficult for FOA to jump away from the local extreme. In order to escape from the local extreme, the fitness function must be corrected (namely, Si must be corrected). In addition, distance Disti is positive in original FOA, and Si is also positive as the reciprocal of distance. Therefore, the FOA fitness function cannot be negative. To avoid above deficiencies, Pan [22] proposed a FOA correction method called MFOA by adding an escape parameter to Si . By using this parameter, the global optimal could be found while escaping from the local optimal, and the fitness function could be negative. The modified Si is expressed as the following equation.

Si ¼ 1=Disti þ D;

D ¼ Disti  ð0:5  dÞ;

06d61

ð12Þ

In addition, since fruit flies move in three-dimensional space, they usually have a very large search space, and it is very easy for them to find food. However, the original FOA searches for global optimal in two-dimensional space, it could possibly lead to difficulties in searching for the global extreme in three-dimensional space. Therefore, the distance of the food location to the origin is extended to three-dimensional space and it is corrected as the following equation.

Disti ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðX 2i þ Y 2i þ Z 2i Þ

ð13Þ

In the MFOA, the escape parameter of d is introduced to be able to generate a negative fitness and enhance FOA’s global search ability. However, the relationship between d and smell concentration judgment value Si and the mapping between location ðX i ; Y i ; Z i Þ and Si are not analyzed. According to Eq. (12), we can see that there is a nonlinear transformation between the fruit fly’s location ðX i ; Y i ; Z i Þ and Si . Obviously, the choice of d has great influence to the optimization solution and is problem depend. So, how to choose the d is a new problem for the MFOA. 2.3. Improved fruit fly optimization algorithm (IFFO) The basic FOA generates food sources around its swarm location within a fixed search radius in the osphresis search phase. To improve the performance of the FOA and eliminate the drawbacks lays with fixed values of search radius, an improved fruit fly optimization algorithm (IFFO) is proposed in [23]. In IFFO, the search radius is changed dynamically with iterations as the following.

    kmin t  k ¼ kmax  exp ln T kmax

ð14Þ

where k is the search radius in each iteration, kmax is the maximum search radius, kmin is the minimum search radius. In IFFO, to avoid the disadvantage of fitness function of FOA, both the distance Disti and smell concentration judgment value Si are removed, and the fitness function is evaluated by the decision variables directly. Moreover, to enhance the intensive search, the IFFO does not change all the decision variants of the swarm location when producing a new position. Instead, only one decision variant is chosen randomly to generate a new solution.

 X ij ¼

X axisj  k  rand; if j ¼ d X axisj ;

otherwise

;

j ¼ 1; 2; . . . ; n

ð15Þ

where the rand is a function which returns a value from the uniform distribution on the interval [0, 1], d is a random integer in the range of [1, n], and n is the number of decision variables. By introducing a new control parameter and an effective solution generating method, the IFFO is testified as a powerful search algorithm to solve the high-dimensional continuous function optimization problems. However, just as the authors mentioned in [23], the IFFO generates small gaps with the true optimal solutions for most of the chosen benchmark test functions. There is room for the IFFO to be further improved. 2.4. Multi-swarm fruit fly optimization algorithm (MSFOA) In order to enhance the diversity of solutions and achieve an effective exploration to avoid local optimal or premature, Yuan et al. [18] proposed a multi-swarm fruit fly optimization (MSFOA). In MSFOA, the swarm is split into several sub-swarms (usually 4–10) and the sub-swarms move independently in the search space with the aim of simultaneously exploring global optimal. Similar to the IFFO in [23], the distance Disti and smell concentration judgment value Si are removed in MSFOA, and the fitness

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ð16Þ

ating method. The motivation is based on the fact that the fruit flies have randomness and fuzziness when they fly to the next possible food source. While cloud model is an effective tool that is able to synthetically describe the randomness and fuzziness. In the following, the normal cloud model will be introduced firstly, and then the normal cloud model based FOA is presented in detail.

ð17Þ

3.1. Normal cloud model

function is evaluated by the decision variables directly. Moreover, the multi-scale exploring radius for osphresis is employed in MSFOA. For each fruit fly, its new location is generated according to the following equations.

X i ¼ X axisj þ RðtÞ  RandomValue; /    UBj  LBj T t  RðtÞ ¼ T 2

j ¼ 1; 2; . . . ; n

where RðtÞ is the exploring radius using osphresis, and this variable is set as a multi-scale factor according to the iteration t; / ¼ 2  6. However, MSFOA has no any information exchanging or sharing mechanism between the sub-swarms, and the improvement of exploration ability for multi-modal problems is limited. Moreover, the MSFOA is only evaluated by 5 low-dimension benchmark functions in [18] and the results should be further verified by more test cases. 3. Cloud model based FOA This section presents an improved fruit fly optimization algorithm by introducing a normal cloud model based solution gener-

Fig. 2. Pseudo-code of normal cloud generator.

Randomness and fuzziness are the two most important uncertainties inherent in the nature, which have attracted great attention in artificial intelligence research. Regarding the shortcomings of fuzzy mathematics and probability theory when handling uncertainty, the concept of cloud model was proposed in 1995 [24]. Cloud models present a new approach to describe the randomness and fuzziness of concepts and implement the uncertain transformation between a qualitative concept and its quantitative instantiations [25]. The cloud model can effectively integrate the randomness and fuzziness of concepts and describe the overall quantitative property of a concept by the three numerical characteristics of Ex (Expectation), En (Entropy) and He (Hyper Entropy) [25]. Expectation Ex is the mathematical expectation of the cloud drops belonging to a concept in the universal. It can be regarded as the most representative and typical sample of the qualitative concept. Entropy En represents the uncertainty measurement of a qualitative concept. It is determined by both the randomness and the fuzziness of the concept. In one aspect, as the measurement of randomness, En reflects the dispersing extent of the cloud drops and in the other hand, it is also the measurement of fuzziness, representing the scope of the universe that can be accepted by the concept. Hyper Entropy He is the uncertain degree of entropy En.

Fig. 3. Clouds generated by normal cloud generator.

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If both the Entropy En and Hyper Entropy He are zero, the qualitative concept is deterministic. Normal cloud model is an important kind of cloud model based on normal distribution and Gauss membership function. The reason is obvious that normal distribution is an important kind of probability distribution in the nature, and has been widely used in natural science and social science. As a specific kind of cloud model, normal cloud model also has the three numerical characteristics and its definition can be described as the following [25].

ness and fuzziness. In addition, the coverage and discrete degree of these clouds has obvious difference that is the larger the Entropy (En), the larger the distribution range; the larger the Hyper Entropy (He), the bigger the discrete degree. Thus, through the procedure of the cloud drops generated by normal cloud generator based on three parameters (Ex; En; He) can describe the foraging behaviors of some biological swarms in the nature. The Ex stands for possible food position, the En represents search range, and the He indicates stability of search.

Definition 1. let U be the universe of discourse and C is a qualitative concept associated with U. If x 2 U is a random instantiation of concept C, which satisfies normal distribution x  NðEx; En0 Þ; En0  NðEn; HeÞ and the certainty degree of x belong-

3.2. Fruit fly position update based on normal cloud generator

2

ing to concept C satisfies y ¼ exp½ðx  ExÞ2 =ð2ðEn0 Þ Þ, then the x is called as cloud drop and the distribution of x in the universe U is called as a normal cloud. Give the three numerical characteristics: Ex; En; He and the number of cloud drops N, the generator of a normal cloud can be described as the algorithm show in Fig. 2. To be convenient, the procedure of the cloud drops generated is defined as following form:

X ¼ CxðEx; En; HeÞ

ð18Þ

Fig. 3 shows clouds generated by normal cloud generator in different parameters. It can be seen that the four clouds are similar to normal distribution, however the thickness of the cloud is uneven which can reflects the features of normal cloud model: random-

When the fruit fly swarm learns to the best individual to fly to the next potential food location, different fruit flies generally have different judgments and their fly directions and routines are various. In other words, the foraging behavior of fruit fly swarm has the features of randomness and fuzziness. Here, this kind of randomness and fuzziness is described by the normal cloud model. The normal cloud generator instead of uniformly distributed random function is used to yield new locations for each fruit fly during the osphresis search phase.

 X ij ¼

CxðX axisj ; En; HeÞ; if j ¼ d X axisj ;

otherwise

;

j ¼ 1; 2; . . . ; n

ð19Þ

where d is a random integer in the range of [1, n]. In Eq. (19), the En is used to represent the search radius, and the He is the entropy of En which stands for the stability of the search. In the early search stage, the fruit fly swarm location is often far away from an optimum solution, and a large search range is needed. However, with the evolution of the swarm, its location is close to an optimum or a near optimum solution, so a small search radius is appropriate to fine tune the solution. Thus, to improve the performance of the FOA, an adaptive strategy of En and He that dynamically varies with iterations is adopted as the follows.

 a t En ¼ En max  1  T

ð20Þ

He ¼ 0:1En

ð21Þ

where En max ¼ ðUB  LBÞ=4 is the maximum search radius, a is a positive integer and it defines the exploitation accuracy over the iterations. The larger a, the sooner and more accurate exploitation. To have a good balance between the exploration and exploitation, a ¼ 10 is used in this paper. 3.3. Procedure of CMFOA The pseudocodes of the proposed CMFOA are outlined in the Algorithm 3 (see Fig. 4). It is worth mentioning here that, similar to the IFFO in [23] and MSFOA in [18], the distance Dist i and smell concentration judgment value Si in original FOA are also removed to avoid the disadvantage of fitness function of FOA. From Algorithm 3 we can see that the main difference between CMFOA and IFFO is the generating method of food source in the osphresis foraging phase. But in the IFFO, the intrinsic fuzziness of fruit fly is not considered, while the randomness and fuzziness are both considered in the CMFOA. 3.4. Computational complexity analysis

Fig. 4. Pseudo-code of CMFOA.

The computational complexity of intelligent optimization algorithm mainly includes two parts: algorithm execution and fitness evaluation. Because different algorithms have the same computational complexity in fitness evaluation, so we just discuss the computational complexity of algorithm execution.

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According to the steps of the algorithm of FOA, the computational complexity of FOA mainly includes the individual position update, concentration value calculation and fitness value evaluation. Assume that the population size is NP and the maximum number of iterations is T. The computational complexity of the individual position update and the concentration value calculation are both Oð2  NP  TÞ. So, the computational complexity of FOA is Oð4  NP  TÞ. For MFOA, because an escape parameter of d is introduced, an extra computational complexity of OðNP  T þ TÞ is increased compared with FOA. So, the computational complexity of MFOA is Oð5  NP  T þ TÞ. For MSFOA, the computational complexity mainly includes the calculation of exploring radius RðtÞ and the individual position update. The computational complexity of calculate RðtÞ is OðTÞ and the computational complexity of individual position update is OðNP  TÞ. So, the computational complexity of MSFOA is OðNP  T þ TÞ.

Table 1 Unimodal benchmark functions. Function P 2 F 1 ðxÞ ¼ ni¼2 ixi P 2 F 2 ðxÞ ¼ ni¼2 ið2x2i  xi1 Þ þ ðx1  1Þ2 Pn 2 F 3 ðxÞ ¼ expð0:5 i¼1 xi Þ P ði1Þ=ðniÞ x2i F 4 ðxÞ ¼ ni¼1 ð106 Þ P 4 F 5 ðxÞ ¼ ni¼1 ixi þ randðÞ P 2 F 6 ðxÞ ¼ ni¼1 ½100ðxiþ1  x2i Þ þ ðxi  1Þ2  Pn Pi 2 F 7 ðxÞ ¼ i¼1 ð j¼1 xj Þ F 8 ðxÞ ¼ maxfjxi j; 1 6 i 6 ng P Q F 9 ðxÞ ¼ ni¼1 jxi j þ ni¼1 jxi j P F 10 ðxÞ ¼ ni¼1 x2i P F 11 ðxÞ ¼ ni¼1 ðbxi þ 0:5cÞ2 P F 12 ðxÞ ¼ ni¼1 jxi jiþ1 P F 13 ðxÞ ¼ ni¼1 ix2i

Range

Fðx Þ

[5.12, 5.12]

0

[10, 10]

0

[1, 1]

1

[100, 100]

0

[1.28, 1.28]

0

[30, 30]

0

[100, 100]

0

[100, 100] [10, 10] [100, 100]

0 0 0

[100, 100]

0

[1, 1]

0

[10, 10]

0

For IFFO, the computational complexity mainly includes calculation of the search radius k and the individual position update. The computational complexity of k calculation is OðTÞ and the computational complexity of individual position update is OðNP  TÞ. So, the computational complexity of IFFO is OðNP  T þ TÞ. For CMFOA, the computational complexity mainly includes calculation of En; He and implementation of normal cloud model. The computational complexity of En and He calculation is Oð2  TÞ and the computational complexity of normal cloud model is Oð2  NP  TÞ. Hence, the computational complexity of FOA is Oð2ðNP  T þ TÞÞ. From the above analysis, we can find that the computational complexity of CMFOA is less than FOA and MFOA and a little larger than MSFOA and IFFO, but the computational complexity of CMFOA is acceptable.

4. Numerical analysis In order to verify the performance of our approach, in this section, CMFOA is applied to 33 scalable benchmark functions with dimensions equal to 30 and 50. The specific description of these functions is shown in Tables 1–3. These benchmark functions are widely adopted in benchmarking global optimization algorithm. Among these test functions, the first 13 unimodal test functions F1-F13 belong to the first group, the second group includes 14 multi-modal functions F14-F27, and the third group includes 6 composite functions F28-F33 provided by CEC2005 special session[26]. To compare the results of different algorithms, each function was optimized over 30 independent runs. To be fair, we used the same set of initial random populations to evaluate different algorithms, and all of the compared algorithms were started from the same initial population in each out of 30 runs. The error value f ðxÞ  f ðx Þ was recorded for the solution x, where f ðxÞ is the minimum values of the function calculated by algorithm, f ðx Þ is the true global minimum of the function. The average and standard deviation of the error values over all independent runs were calculated. The results are summarized as ‘‘w=t=l”, which means that

Table 2 Multi-modal benchmark functions. Function ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P F 14 ðxÞ ¼ 20 expð0:2 1n ni¼1 x2i Þ  expð1n ni¼1 cosð2pxi ÞÞ þ 20 þ expð1Þ Pn F 15 ðxÞ ¼ i¼1 jxi sinðxi Þ þ 0:1xi j F 16 ðxÞ ¼ f 10 ðx1 ; x2 Þ þ    þ f 10 ðxn ; x1 Þ; 0:25

0:1

½sin2 ð50ðx2 þ y2 Þ Þ þ 1 f s ðx; yÞ ¼ ðx2 þ y2 Þ F 17 ðxÞ ¼ f s ðxi ; x2 Þ þ    þ f s ðxn ; x1 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f s ðxÞ ¼ 0:5 þ ðsin2 ð x2 þ y2 Þ  0:5Þ=ð1 þ 0:001ðx2 þ y2 ÞÞ P P F 18 ðxÞ ¼ pn f10 sin2 ðpyi Þ þ ni¼1 ðyi  1Þ2 ½1 þ 10 sin2 ðpyiþ1 Þ þ ðyn  1Þ2 þ ni¼1 uðxi ; 10; 100; 4Þ; 8 m xi > a < kðxi  aÞ ; yi ¼ 1 þ 14 ðxi þ 1Þ; uðxi ; a; k; mÞ ¼ 0; a 6 xi 6 a : m kðxi  aÞ ; xi < a pffi Q P n n 1 2 F 19 ðxÞ ¼ 4000 i¼1 xi  i¼1 cosðxi = iÞ þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 F 20 ¼  n1 ðexpððx þ x2iþ1 þ 0:5xi xiþ1 Þ=8Þ  cosð4 x2i þ x2iþ1 þ 0:5xi xiþ1 ÞÞ i¼1 i P P F 21 ðxÞ ¼ ni¼1 ðxi  1Þ2  ni¼2 xi xi1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn1 2 F 22 ðxÞ ¼ i¼1 ð0:5 þ ðsin2 ð 100x2i þ x2iþ1 Þ  0:5ÞÞ=ð1 þ 0:001ðx2i  2xi xi1 þ x2i1 ÞÞ Pn F 23 ðxÞ ¼ i¼1 ½x2i  10 cosð2pxi Þ þ 10 P F 24 ðxÞ ¼ ni¼1 ðy2i  10 cosð2pyi Þ þ 10Þ;  jxi j < 0:5 xi yi ¼ roundð2xi Þ=2 jxi j P 0:5 qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi Pn 2 Pn 2 F 25 ðxÞ ¼ 1  cosð2p i¼1 xi Þ þ 0:1 i¼1 xi Pn Pkmax k P max k k k ½a cosð2pb 0:5Þ F 26 ðxÞ ¼ i¼1 f k¼0 ½a cosð2pb ðxi þ 0:5ÞÞg  n kk¼0 Pn Pn 2 F 27 ðxÞ ¼ k¼1 j¼1 ððyjk =4000Þ  cosðyjk Þ þ 1Þ; 2

yjk ¼ 100ðxk  x2j Þ þ ð1  x2j Þ

2

Range

Fðx Þ

[32, 23]

0

[10, 10]

0

[100, 100]

0

[100, 100]

0

[50, 50]

0

[100, 100]

0

[5, 5]

1n

[n2 ; n2 ]

nðnþ4Þðn1Þ 6

[100, 100]

0

[5.12, 5.12]

0

[5.12, 5.12]

0

[100, 100]

0

[0.5, 0.5]

0

[100, 100]

0

Please cite this article in press as: L. Wu et al., A cloud model based fruit fly optimization algorithm, Knowl. Based Syst. (2015), http://dx.doi.org/10.1016/j. knosys.2015.09.006

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L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx

CMFOA wins in w functions, ties in t functions, and loses in l functions, compared with its competitors. 4.1. Parameter setting For all algorithms in this paper, the population size NP ¼ 50 and the maximal number of fitness function evaluations NFESmax ¼ 100000 were used. For FOA and MFOA, set the randValue ¼ 1 as in [9,22]. For MSFOA, M ¼ 5 and / ¼ 6 were used as in [18]. k max ¼ ðUB  LBÞ=2 and k min ¼ 1  105 were used for IFFO as in [23]. The parameters of CMFOA are the same as the previous. The choice of the control parameters of CMFOA are mainly referenced IFFO and MSFOA and based on our extensive experiments. 4.2. Comparison among different FOA variants In this section, CMFOA is compared with three recent improved FOA algorithms MFOA [22], MSFOA [18], IFFO [23], and the basic

FOA. The error values (Error) and standard deviations (StdDev) of CMFOA, IFFO, MSFOA, MFOA and FOA are shown in Tables 4–9 for all functions at D ¼ 30 and D ¼ 50, respectively. In order to highlight the overall best results, the significantly better values are marked in bold. Unimodal test functions have one global optimum, and they are suitable for benchmarking the exploitation ability of algorithms [7]. As we can see from Tables 4 and 5, for most of unimodal benchmark problems the CMFOA has better convergence precision than other four FOA variants chosen to be compared in this paper, which testifies that the proposed algorithm has good exploitation ability. This is due to the parameter adaptive mechanism of En and He, which allows the randomness and fuzziness of fruit flies are decreased over the iterations to have more precise exploitation around the best food source position obtained. In contrast to the unimodal functions, multi-modal functions have a global optimum as well as many local optima with the number increasing exponentially with dimension, which makes them suitable for testing the exploration ability of an algorithm [6].

Table 3 Composite benchmark functions. Function P F 28 ðxÞ ¼ ni¼1 z2i þ f bias P P 2 F 29 ðxÞ ¼ ni¼1 ð ij¼1 zj Þ þ f bias Pn1 2 F 30 ðxÞ ¼ i¼1 ð100ðz2i  ziþ1 Þ þ ðzi  1Þ2 Þ þ f bias pffi Pn 2 Qn 1 F 31 ðxÞ ¼ 4000 i¼1 xi  i¼1 cosðxi = iÞ þ 1 þ f bias Pn F 32 ðxÞ ¼ i¼1 ½x2i  10 cosð2pxi Þ þ 10 F 33 ðxÞ ¼ F 7 ðF 6 ðx1 ; x2 ÞÞ þ F 7 ðF 6 ðx2 ; x3 ÞÞ þ    þ F 7 ðF 6 ðxn ; x1 ÞÞ þ f bias

Fðx Þ

Range [100, 100]

450

[100, 100]

450

[100, 100]

390

[0, 600]

180

[5, 5]

330

[3, 1]

130

Table 4 Comparison results among different FOA variants for unimodal functions at D ¼ 30. Fun

CMFOA

IFFO

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 w=t=l

1.42E24 ± 1.40E24 7.62E01 ± 5.08E01 6.00E16 ± 1.19E16 4.71E18 ± 6.57E18 2.88E02 ± 1.19E02 5.63E+01 ± 3.70E+01 1.25E21 ± 1.34E21 8.19E07 ± 6.99E07 5.07E12 ± 2.45E12 4.08E23 ± 3.92E23 3.44E23 ± 3.01E23 3.61E27 ± 7.29E27 1.57E23 ± 1.61E23 –

1.70E12 ± 9.16E13 6.76E01 ± 5.13E02 3.91E14 ± 1.35E14 1.43E08 ± 1.71E08 2.81E02 ± 1.41E02 5.19E+01 ± 3.63E+01 2.38E12 ± 7.79E13 1.61E06 ± 2.72E07 1.56E06 ± 2.65E07 1.57E13 ± 6.58E14 1.72E13 ± 6.60E14 1.49E15 ± 1.60E15 1.79E12 ± 6.19E13 10/3/0

MSFOA + = + + = = + + + + + + +

7.44E08 ± 1.68E07 1.06E+01 ± 2.78E+01 2.40E15 ± 2.35E15 2.48E+07 ± 1.15E+07 3.95E02 ± 2.05E02 2.64E+02 ± 7.89E+02 2.07E01 ± 5.69E01 8.15E01 ± 9.02E01 5.87E+24 ± 3.20E+25 3.52E11 ± 2.72E11 4.22E11 ± 4.47E11 3.04E07 ± 1.40E07 3.26E03 ± 6.50E03 12/1/0

MFOA + + + + = + + + + + + + +

5.88E05 ± 7.51E06 6.69E01 ± 7.51E03 1.99E06 ± 2.39E07 2.13E+00 ± 5.37E01 2.25E02 ± 6.45E03 1.12E+01 ± 1.31E+00 8.00E05 ± 2.14E05 8.80E04 ± 8.21E05 1.87E02 ± 9.36E03 4.10E06 ± 4.12E07 1.04E04 ± 1.05E05 1.37E11 ± 7.59E12 6.87E05 ± 1.09E05 10/3/0

FOA + = + + = = + + + + + + +

1.82E03 ± 1.69E05 8.72E01 ± 9.11E02 6.70E05 ± 8.52E07 2.79E+00 ± 7.92E02 2.84E02 ± 4.42E03 2.71E+01 ± 1.43E01 1.43E03 ± 3.55E05 5.78E03 ± 2.99E04 5.52E02 ± 5.32E04 1.02E04 ± 1.91E06 7.56E+00 ± 5.52E04 2.05E07 ± 1.11E09 1.83E03 ± 1.83E05 10/3/0

+ = + + = = + + + + + + +

Table 5 Comparison results among different FOA variants for unimodal functions at D ¼ 50. Fun

CMFOA

IFFO

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 w=t=l

1.43E19 ± 1.19E19 3.70E+00 ± 3.30E+00 1.08E15 ± 1.90E16 1.66E13 ± 1.34E13 8.26E02 ± 2.74E02 1.45E+02 ± 2.88E+02 2.06E20 ± 2.42E20 6.30E03 ± 5.34E03 1.58E09 ± 7.01E10 2.24E18 ± 2.22E18 1.57E18 ± 1.06E18 3.10E26 ± 3.71E26 8.61E19 ± 6.45E19 –

1.25E11 ± 4.15E12 3.26E+00 ± 2.87E+00 2.13E13 ± 4.44E14 6.73E08 ± 4.76E08 9.58E02 ± 4.05E02 1.25E+02 ± 2.16E+02 6.96E12 ± 3.45E12 2.13E+00 ± 3.51E+00 4.65E06 ± 6.18E07 8.91E13 ± 2.55E13 9.42E13 ± 2.78E13 5.34E15 ± 6.99E15 1.62E11 ± 7.31E12 10/3/0

MSFOA + = + + = + + + + + + + +

3.11E02 ± 6.12E02 2.96E+01 ± 4.59E+01 2.33E12 ± 1.97E12 6.58E+07 ± 2.81E+07 1.47E01 ± 5.90E02 1.09E+03 ± 2.20E+03 2.73E01 ± 3.79E01 2.35E+01 ± 7.98E+00 7.31E+47 ± 3.79E+48 3.96E08 ± 2.99E08 4.28E08 ± 2.76E08 4.50E07 ± 2.27E07 1.09E+01 ± 2.39E+01 13/0/0

MFOA + + + + + + + + + + + + +

3.27E04 ± 4.50E05 6.69E01 ± 2.21E03 6.30E06 ± 5.86E07 6.24E+00 ± 1.22E+00 4.28E02 ± 8.70E03 3.26E+01 ± 3.77E+00 8.13E05 ± 1.60E05 1.51E02 ± 4.65E03 4.83E02 ± 2.63E02 1.22E05 ± 9.97E07 3.13E04 ± 2.87E05 1.24E11 ± 4.66E12 3.76E04 ± 4.81E05 10/1/2

FOA +  + + =  + + + + + + +

8.32E03 ± 7.71E05 1.33E+00 ± 3.54E01 1.86E04 ± 1.74E06 6.81E+00 ± 1.49E01 6.03E02 ± 1.24E02 4.72E+01 ± 2.87E01 1.44E03 ± 4.12E05 8.59E03 ± 5.23E04 1.14E01 ± 1.28E03 2.64E04 ± 5.62E06 1.26E+01 ± 1.40E03 2.09E07 ± 3.04E09 8.24E03 ± 8.88E05 10/2/1

+ = + + =  + + + + + + +

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L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx

The results of multi-modal test functions with D ¼ 30 and D ¼ 50 can be seen in Tables 6 and 7. Observed the statistic results in Tables 6 and 7, it can be seen that the CMFOA is also able to provide very promising performance. The competitive results indicate that

the CMFOA has better global convergence ability compared with original FOA, MFOA, IFFO and MSFOA. This is mainly contributed by the normal cloud model based osphresis foraging technique, which can not only describe the randomness but also the fuzziness

Table 6 Comparison results among different FOA variants for multi-modal functions at D ¼ 30. Fun

CMFOA

IFFO

F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26 F27 w=t=l

1.34E12 ± 5.80E13 8.53E14 ± 4.82E14 1.71E05 ± 4.25E06 1.10E+00 ± 3.63E01 7.81E26 ± 1.11E25 9.43E03 ± 1.11E02 5.68E+00 ± 1.26E+00 3.03E+03 ± 4.15E+03 1.51E+00 ± 3.48E01 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 1.45E+00 ± 3.16E01 1.34E08 ± 1.34E08 1.81E+01 ± 3.44E+01 –

2.74E07 ± 5.93E08 1.09E06 ± 7.92E07 1.50E02 ± 1.16E03 1.26E+00 ± 5.32E01 2.68E15 ± 2.33E15 9.19E03 ± 1.01E02 5.90E+00 ± 1.68E+00 6.25E+03 ± 5.44E+03 1.75E+00 ± 3.29E01 1.95E11 ± 7.01E12 1.98E11 ± 7.72E12 1.40E+00 ± 3.00E01 3.05E03 ± 4.35E04 1.81E+01 ± 4.00E+01 8/6/0

MSFOA + + + = + = = + = + + = + =

9.20E01 ± 8.38E01 7.85E+00 ± 3.91E+00 2.04E+02 ± 2.33E+01 1.29E+01 ± 4.75E01 1.60E+00 ± 1.36E+00 1.55E02 ± 1.61E02 2.20E+01 ± 1.41E+00 7.59E+03 ± 6.53E+03 5.91E+00 ± 2.31E01 1.48E+02 ± 4.54E+01 1.91E+02 ± 5.17E+01 6.71E01 ± 9.43E02 1.45E+01 ± 2.88E+00 6.07E+02 ± 6.99E+01 12/2/0

MFOA + + + + + + + + = + +  +

1.47E03 ± 6.73E05 8.88E04 ± 6.13E05 6.98E+00 ± 2.29E01 8.23E06 ± 9.37E07 6.32E01 ± 7.65E01 2.10E07 ± 2.78E08 5.28E01 ± 1.49E+00 5.05E+04 ± 5.29E+03 4.62E08 ± 1.38E08 5.97E+00 ± 1.18E+01 1.31E+00 ± 1.16E+00 2.86E04 ± 1.92E05 4.55E+01 ± 2.02E+00 4.14E+02 ± 1.20E03 9/0/5

FOA + + +  +   +  + +  + +

8.30E03 ± 5.68E05 7.18E02 ± 1.70E01 4.71E+00 ± 4.59E01 2.04E04 ± 5.21E06 1.68E+00 ± 6.55E05 5.31E06 ± 1.19E07 6.22E+00 ± 4.24E+00 4.96E+03 ± 5.74E01 6.35E04 ± 2.38E03 6.81E+01 ± 2.95E+01 2.55E+01 ± 3.40E+01 3.03E03 ± 5.19E05 4.45E+01 ± 1.57E+00 4.13E+02 ± 2.27E+00 8/4/2

+ + +  +  = =  + +  + +

Table 7 Comparison results among different FOA variants for multi-modal functions at D ¼ 50. Fun

CMFOA

IFFO

F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26 F27 w=t=l

2.34E10 ± 9.80E11 5.97E11 ± 7.06E11 1.72E01 ± 5.34E01 1.99E+00 ± 5.60E01 2.64E21 ± 1.86E21 7.39E03 ± 7.92E03 1.04E+01 ± 1.96E+00 3.71E+04 ± 2.68E+04 2.97E+00 ± 4.23E01 3.31E02 ± 3.04E01 3.33E02 ± 1.83E01 2.25E+00 ± 3.74E01 1.36E06 ± 3.44E07 1.15E+02 ± 2.00E+02 –

4.90E07 ± 7.86E08 3.90E06 ± 1.80E06 5.21E01 ± 2.32E+00 1.98E+00 ± 6.32E01 6.79E15 ± 4.80E15 1.52E02 ± 2.65E02 1.04E+01 ± 1.54E+00 7.14E+04 ± 5.34E+04 3.26E+00 ± 5.36E01 9.96E01 ± 2.69E01 1.67E01 ± 3.79E01 2.23E+00 ± 4.87E01 7.32E03 ± 6.25E04 1.29E+02 ± 2.37E+02 8/6/0

MSFOA + + + = + + = = = + + = + =

4.95E+00 ± 6.14E+00 1.57E+01 ± 4.17E+00 3.81E+02 ± 2.92E+01 2.22E+01 ± 7.08E01 1.03E+02 ± 4.28E+02 5.58E03 ± 9.16E03 3.90E+01 ± 2.21E+00 1.41E+05 ± 1.41E+05 1.07E+01 ± 2.78E01 3.29E+02 ± 7.92E+01 4.07E+02 ± 6.63E+01 1.55E+00 ± 2.38E01 3.62E+01 ± 4.29E+00 1.86E+03 ± 1.25E+02 11/3/0

MFOA + + + + + = = + + + + = + +

2.00E03 ± 1.08E04 3.40E02 ± 1.75E01 1.17E+01 ± 4.12E01 2.53E05 ± 1.92E06 1.03E+00 ± 8.07E01 4.76E07 ± 6.25E08 1.01E+00 ± 1.99E+00 2.10E+06 ± 5.13E+05 3.12E07 ± 1.64E07 1.98E+01 ± 2.97E+01 3.18E+00 ± 4.37E+00 6.02E04 ± 3.82E05 8.13E+01 ± 1.90E+00 1.15E+03 ± 4.38E03 9/0/5

FOA + + +  +   +  + +  + +

1.07E02 ± 5.59E05 4.49E02 ± 1.71E01 7.87E+00 ± 7.02E01 5.32E04 ± 1.03E05 1.48E+00 ± 5.64E05 8.84E06 ± 2.16E07 1.54E+01 ± 7.16E+00 2.21E+04 ± 4.28E01 1.40E03 ± 3.60E03 1.27E+02 ± 4.80E+01 1.24E+02 ± 7.25E+01 9.92E03 ± 1.70E02 7.96E+01 ± 1.92E+00 1.15E+03 ± 4.65E+00 8/2/4

+ + +  +  = =  + +  + +

Table 8 Comparison results among different FOA variants for composite functions at D ¼ 30. Fun

CMFOA

IFFO

F28 F29 F30 F31 F32 F33 w=t=l

3.18E13 ± 8.39E14 9.88E+02 ± 6.29E+02 9.70E+01 ± 6.02E+01 1.58E02 ± 1.28E02 3.28E13 ± 6.79E14 1.22E+00 ± 2.97E01 –

3.53E13 ± 6.63E13 4.57E+03 ± 1.54E+03 5.73E+02 ± 1.68E+03 5.17E02 ± 2.16E01 2.99E13 ± 8.17E14 1.24E+00 ± 2.64E01 2/4/0

MSFOA = + + = = =

2.95E11 ± 2.55E11 7.11E+0 ± 6.46E+0 1.82E+05 ± 9.40E+05 1.22E02 ± 1.07E02 1.48E+02 ± 3.21E+01 7.56E+00 ± 2.37E+00 3/2/1

MFOA +  + = = =

8.83E+04 ± 1.62E+04 3.32E+06 ± 2.51E+06 4.43E+10 ± 8.14E+09 2.63E+00 ± 2.76E+00 4.32E+02 ± 3.75E+01 2.66E+01 ± 2.36E+00 6/0/0

FOA + + + + + +

8.60E+04 ± 2.75E+03 1.16E+06 ± 3.14E+00 4.20E+10 ± 1.24E+09 4.79E+03 ± 4.49E+01 3.62E+02 ± 1.79E+01 1.96E+02 ± 1.00E+01 6/0/0

+ + + + + +

Table 9 Comparison results among different FOA variants for composite functions at D ¼ 50. Fun

CMFOA

IFFO

F28 F29 F30 F31 F32 F33 w=t=l

5.76E13 ± 7.57E14 1.55E+04 ± 4.24E+03 1.43E+03 ± 3.31E+03 1.50E02 ± 8.66E03 3.32E02 ± 1.82E01 2.24E+00 ± 4.56E01 –

1.14E12 ± 2.01E13 2.05E+04 ± 5.23E+03 1.13E+03 ± 3.06E+03 4.65E+02 ± 1.09E+02 5.64E01 ± 7.24E01 2.71E+00 ± 5.08E01 3/3/0

MSFOA + = = + + =

4.89E08 ± 4.65E08 9.57E+04 ± 3.51E+04 6.20E+04 ± 1.90E+05 4.06E02 ± 2.94E02 3.56E+02 ± 8.46E+01 1.80E+01 ± 4.57E+00 4/2/0

MFOA + + = + + =

1.49E+05 ± 2.33E+04 1.55E+07 ± 8.94E+06 7.45E+10 ± 1.45E+10 1.29E+02 ± 2.55E+02 7.83E+02 ± 4.79E+01 6.44E+01 ± 6.25E+00 6/0/0

FOA + + + + + +

1.44E+05 ± 3.14E+03 5.78E+06 ± 2.36E+01 6.44E+10 ± 1.31E+09 6.47E+03 ± 4.57E+01 6.99E+02 ± 2.69E+01 6.97E+02 ± 4.61E+01 6/0/0

+ + + + + +

Please cite this article in press as: L. Wu et al., A cloud model based fruit fly optimization algorithm, Knowl. Based Syst. (2015), http://dx.doi.org/10.1016/j. knosys.2015.09.006

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L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx

5

0

10

10

10

10

0

5

10

−10

10

CMFOA

−15

10

IFFO

CMFOA 10

0

2

4

6

8

NEFS

10

2

4

4

6

8

10

10

CMFOA IFFO

10

−6

10

0

CMFOA IFFO

−20

10

FOA MSFOA

−30

6

8

10

0

2

6

8

NEFS

(d) F16 Average convergence curve 5

10

10

CMFOA IFFO FOA

−15

10

−20

0

2

−4

CMFOA

10

IFFO FOA

4

6

8

NEFS

10

MSFOA 0

2

(g) F24 Average convergence curve

6

8

NEFS

10

10

12

MFOA MSFOA 0

2

IFFO

10 Mean Errors

MFOA MSFOA

10

4

10

6

8

10 4

x 10

0

FOA

10

MFOA MSFOA

2

10

0

CMFOA IFFO

−10

10

2

−5

10

10

FOA MFOA

10

0

10

4

5

4

FOA

6

FOA

CMFOA

IFFO

10

IFFO

10

CMFOA

8

4

(i) F28 Average convergence curve

6

10

10 x 10

NEFS

10

10

8

CMFOA

4

x 10

(h) F26 Average convergence curve

10

−5

10

−15

4

4

x 10

6

0

10

MFOA

−8

10

4

10

−10

Mean Errors

10

−2

10

10

MSFOA

2

5

−6

MFOA

0

10 Mean Errors

Mean Errors

−10

10

MSFOA

10

0

−5

MFOA

10

10

10

FOA

(f) F23 Average convergence curve

2

0

IFFO

NEFS

10

10

CMFOA

4

x 10

(e) F18 Average convergence curve

10

−10

10

−20

4

4

x 10

4

−5

10

MSFOA

NEFS

10 x 10

10

−15

MFOA

4

8

0

MFOA

2

6

10

10

10

4

5

−10

FOA

−4

2

10

Mean Errors

Mean Errors

−2

10

0

(c) F4 Average convergence curve

0

0

MSFOA

NEFS

10

10

MFOA

4

x 10

10

2

FOA

−20

10

10

IFFO

−15

(b) F3 Average convergence curve

10

CMFOA

−10

10 10

NEFS

(a) F1 Average convergence curve

Mean Errors

0

4

x 10

−5

10

MSFOA

−20

10

10

FOA MFOA

MSFOA

−25

Mean Errors

IFFO

−15

MFOA

10 10

−10

10

FOA

−20

0

Mean Errors

Mean Errors

Mean Errors

−5

10

Mean Errors

10

−5

10

−2

0

2

4

6 NEFS

8

10

10

0

2

(j) F30 Average convergence curve

4

6 NEFS

4

x 10

MSFOA

−15

8

10 4

x 10

(k) F31 Average convergence curve

10

0

2

4

6 NEFS

8

10 4

x 10

(l) F32 Average convergence curve

Fig. 5. Average convergence curves over 30 runs of different FOA variants for the selected functions (D ¼ 30).

of the fruit fly swarm foraging behavior. This solution generating technique can help the fruit fly swarm to escape from local optimal.

Next, the balance between exploration and exploitation will be discussed. It is an essential feature of stochastic algorithms for solving real problems [7]. The composite test functions are more

Please cite this article in press as: L. Wu et al., A cloud model based fruit fly optimization algorithm, Knowl. Based Syst. (2015), http://dx.doi.org/10.1016/j. knosys.2015.09.006

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L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx

5

0

10

10

10

10

0

5

10

10

10

−10

CMFOA

10

IFFO

−15

10

0

CMFOA

−10

10

IFFO

FOA

FOA

MFOA

MFOA

MSFOA

−20

10

10

2

4

6

8

NEFS

10

0

2

3

IFFO FOA MFOA MSFOA

−15

4

6

8

10

10

0

2

4

6

4

(c) F4 Average convergence curve

10

3

10

10

10

MFOA MSFOA

10

Mean Errors

FOA

1

2

−10

10

CMFOA

0

IFFO

−20

10

10

FOA

−1

0

2

4

6

8

NEFS

10

2

6

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10

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CMFOA IFFO

10

−2

10

0

2

−2

10

CMFOA IFFO

10

FOA

MSFOA

MSFOA

4

6

8

NEFS

10

0

2

(g) F24 Average convergence curve

6

8

NEFS

12

10

10

2

10

CMFOA IFFO

10

4

10

FOA

2

8

4

10

10

0

2

(j) F30 Average convergence curve

CMFOA

0

10

IFFO FOA MFOA MSFOA

−2

4

6 NEFS

4

x 10

1

10

10

MSFOA

−2

6

10 x 10

10

−1

MFOA

NEFS

8

2

0

4

6

10

Mean Errors

Mean Errors

Mean Errors

MSFOA

2

4

4

10

MFOA

0

2

10

4

FOA

6

10

0

3

IFFO

8

MSFOA

(i) F28 Average convergence curve

CMFOA

10

MFOA

NEFS

6

10

FOA

4

x 10

10

10

IFFO

10

(h) F26 Average convergence curve

10

CMFOA

−15

4

4

x 10

−5

10

10

MFOA −6

0

10

−10

MFOA 10

4

5

−4

FOA

−1

10 x 10

10 Mean Errors

Mean Errors

0

10

8

10

0

1

6

10

10

10

4

(f) F23 Average convergence curve

2

2

2

NEFS

10

10

0

4

x 10

(e) F18 Average convergence curve

3

0

10

−2

4 NEFS

10

MSFOA

−1

4

x 10

(d) F16 Average convergence curve

Mean Errors

0

MFOA

10

MSFOA

−30

10

FOA

1

10

MFOA 10

IFFO

10

0

Mean Errors

Mean Errors

CMFOA

IFFO

10

10 x 10

CMFOA 2

8

NEFS

4

x 10

(b) F3 Average convergence curve

10

CMFOA

10

NEFS

(a) F1 Average convergence curve

−5

10

−10

4

x 10

0

10

MSFOA

−15

10

Mean Errors

−5

Mean Errors

Mean Errors

−5

8

10 4

x 10

(k) F31 Average convergence curve

10

0

2

4

6 NEFS

8

10 4

x 10

(l) F32 Average convergence curve

Fig. 6. Average convergence curves over 30 runs of different FOA variants for the selected functions (D ¼ 50).

similar to the real search spaces and good for examining the balance ability of global exploration and local exploitation. The results of the algorithms on composite test functions are shown in Tables

8 and 9. As presented in Tables 8 and 9, the CMFOA is slightly better than IFFO and MSFOA and significantly better than MFOA and FOA. This evidences that the CMFOA algorithm provides highly

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competitive results on the composite test functions as well and has acceptable global exploration and local exploitation ability. This is mainly due to the parameter adaptive technique of the normal cloud model, which allows the CMFOA has large randomness and fuzziness in the early stage to enhance the global exploration ability and decreases the randomness and fuzziness over the iterations to gradually emphasis the exploitation ability. Therefore, a good balance between the exploration and exploitation is reached. Tables 8 and 9 also clearly show that because of the intrinsic deficiencies of the fitness function definition, FOA and MFOA are not able to solve the composite problems. To testify the above results, some representative convergence graphs of the FOA variants are illustrated in Figs. 5 and 6. From Figs. 5 and 6, it can be observed that for most test problems the CMFOA can converge to the global optimal solution with faster convergence speed and higher accuracy than other four algorithms. Figs. 5 and 6 also clearly show that FOA and MFOA are easy to be trapped to the local optimal because of the inherent deficiencies of the definition of fitness function. While the global convergence ability of CMFOA, IFFO and MSFOA are improved greatly. One of the reasons is that the improper distance Disti and smell concentration judgment value Si in original FOA are discarded. In addition, Tables 4–9 and Figs. 5 and 6 show that CMFOA generally has faster convergence rate and better accuracy than IFFO and MSFOA. This is mainly contributed by the normal cloud model based osphresis foraging technique which can not only describe the randomness but also the fuzziness of the fruit fly swarm foraging behavior. To further testify the above results, we present the results of the Wilcoxon signed-rank test, calculated by the KEEL software tool [27], for all Benchmark functions at D ¼ 30 and D ¼ 50 in Tables 10 and 11, respectively. Where ‘‘ ” means that the method in the row improves the method of the column, and the ‘‘ ” means that the method in the column improves the method of the row. Upper diagonal of level significance a ¼ 0:1, and lower diagonal level of significance a ¼ 0:05. As we can see from Table 10, CMFOA is significantly better than FOA, MFOA, MSFOA, and IFFO at D ¼ 30. For Benchmark functions at D ¼ 50, there are similar results as D ¼ 30.

4.3. Comparison among different intelligent optimization algorithms In this section, CMFOA is compared with seven representative intelligent optimization algorithms CLPSO [28], IASFA [29], SGHS

Table 10 Ranks computed by the Wilcoxon test for FOA variants on Benchmark functions at D ¼ 30.

FOA (1) MFOA (2) MSFOA (3) IFFO (4) CMIFOA (5)

(1)

(2)

(3)

(4)

– 300.0 217.0 451.0

476.0

261.0 – 237.0 478.0

476.0

344.0 324.0 – 508.0

512.0

110.0 83.00 53.00 – 410.0

(5)

85.00 85.00 49.00 118.0 –



Table 11 Ranks computed by the Wilcoxon test for FOA variants on Benchmark functions at D ¼ 50.

FOA (1) MFOA (2) MSFOA (3) IFFO (4) CMFOA (5)

(1)

(2)

(3)

(4)

– 304.0 199.0 417.0

430.0

224.0 – 220.0 420.0

463.0

362.0 341.0 – 519.0

544.0

144.0 141.0 42.00 – 412.0

(5)

131.0 98.00 17.00 116.0 –



11

[30], SaDE [31], GWO [6], MVO [7], ALO [8]. Among them, the CLPSO and SaDE are the Top Access Articles in the PSO and DE community [32], respectively. Moreover, the SaDE is one of the famous DEs which adopt self-adaptive parameters [32] and it won the 2012 IEEE CIS TEC Outstanding Paper Award(http://cis.ieee.org/ award-recipients.html). The parameter adaptive strategy can enhance the robustness of DE by dynamically adapting the parameters to the characteristics of different fitness landscapes [33,34]. The GWO, MVO and ALO are the latest-developed swarm intelligent optimization algorithms. While the IASFA and SGHS are the representatives of other swarm intelligence. The parameters of these algorithms are the same as the corresponding literatures suggested. The parameters of the CMFOA are the same as the previous. The results of the unimodal benchmark functions with D ¼ 30 and D ¼ 50 compared with different intelligent optimization algorithms are presented in Tables 12 and 13. The results show that for most problems F1-F13, the CMFOA has better global convergence ability and convergence precision than the CLPSO, IASFA, SGHS, ALO and MVO. When it is compared with the SaDE and GWO, the two algorithms have better convergence precision than the CMFOA for most unimodal problems. But the CMFOA has competitive exploration ability with the SaDE and GWO. When it comes to the multi-modal benchmark functions, the comparison results are shown in Tables 14 and 15. The results evidence that the CMFOA has better global convergence ability and convergence precision than the CLPSO, IASFA, SGHS, ALO, MVO and SaDE for most multi-modal problems. When compared with the GWO, the CMFOA can get competitive results, that is to say the performances of the CMFOA and GWO are comparable in solve multi-modal problems. The results of the composite benchmark functions with D ¼ 30 and D ¼ 50 are shown in Tables 16 and 17. As we can see from Tables 16 and 17, all the algorithms cannot obtain the global optimal solution for most composite functions, this is due to the difficulty of this set of test functions. However, the CMFOA also can get a highly competitive results on the composite functions compared with other seven different intelligent optimization algorithms. Furthermore, to testify the above results, we present the results of the Wilcoxon signed-rank test for all Benchmark functions at D ¼ 30 and D ¼ 50 in Tables 18 and 19, respectively. Observed Tables 18 and 19, it can be seen that the CMFOA is significantly better than the CLPSO, IASFA and SGHS for all test functions at D ¼ 30 and D ¼ 50. When it compares to the SaDE and GWO, just as the previous analysis, the CMFOA loses in unimodal functions but wins in multi-modal and composite functions. So, there is still room for further improving the exploitation ability of the CMFOA. The superior performances of the proposed CMFOA are mainly benefited from a new solution generating method based on normal cloud generator and a parameter adaptive strategy. Similar to other biological swarms in nature, fruit flies also demonstrate the collective behavior while hunting food. While searching, a fruit fly sends and receives information from its neighbors and makes comparison with the so far best location and fitness. However, different flies generally have different judgements and decisions when they learn to the elites because of the inborn individual discrepancies. That is to say, when fruit flies learn to the best individual and fly to a new promising food source, the foraging behavior of fruit fly swarm has features of randomness and fuzziness. Therefore, the normal cloud model is used to describe the randomness and fuzziness of the foraging behavior of fruit fly swarm. As a result, the population diversity is well maintained and the global search ability is greatly improved. Moreover, a good optimization algorithm should be such one which has the global exploration ability to find as many as possible

Please cite this article in press as: L. Wu et al., A cloud model based fruit fly optimization algorithm, Knowl. Based Syst. (2015), http://dx.doi.org/10.1016/j. knosys.2015.09.006

12

L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx

Table 12 Comparison results among different intelligent algorithms for unimodal functions at D ¼ 30. Fun

CMFOA

CLPSO

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 w=t=l

1.42E24 ± 1.40E24 7.62E01 ± 5.08E01 6.00E16 ± 1.19E16 4.71E18 ± 6.57E18 2.88E02 ± 1.19E02 5.63E+01 ± 3.70E+01 1.25E21 ± 1.34E21 8.19E07 ± 6.99E07 5.07E12 ± 2.45E12 4.08E23 ± 3.92E23 3.44E23 ± 3.01E23 3.61E27 ± 7.29E27 1.57E23 ± 1.61E23 –

4.98E07 ± 1.83E07 2.05E+00 ± 9.46E01 6.09E10 ± 1.82E10 4.47E02 ± 2.00E02 1.75E02 ± 6.55E03 6.82E+01 ± 2.67E+01 2.00E04 ± 6.56E05 1.62E+01 ± 1.54E+00 1.12E02 ± 2.61E03 1.64E05 ± 7.37E06 1.53E05 ± 6.70E06 1.17E28 ± 2.12E28 1.82E06 ± 6.96E07 10/2/1

Fun

SGHS

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 w=t=l

2.95E08 ± 2.57E08 1.01E+00 ± 1.28E+00 0.00E+00 ± 0.00E+00 3.40E+01 ± 6.69E+02 3.57E02 ± 1.85E02 8.67E+01 ± 9.98E+01 1.13E+03 ± 4.36E+02 6.91E01 ± 2.33E01 2.08E04 ± 9.89E04 2.76E04 ± 5.03E03 0.00E+00 ± 0.00E+00 1.02E12 ± 5.13E12 3.41E08 ± 4.55E05 9/2/2

IASFA + + + + = = + + + + +  +

1.62E+01 ± 8.74E+01 4.79E+00 ± 3.29E+00 3.27E01 ± 1.21E01 2.59E+05 ± 2.03E+05 3.72E02 ± 3.41E02 6.59E+01 ± 4.87E+01 4.65E+01 ± 1.55E+02 1.50E+01 ± 1.19E+0+1 3.49E+19 ± 1.91E+20 3.24E+00 ± 5.03E+00 5.10E+01 ± 2.56E+02 2.26E53 ± 4.60E53 1.29E+00 ± 9.67E01 10/2/1

ALO + +  + = = + + + + + + +

8.22E06 ± 7.29E06 7.89E01 ± 2.06E01 2.92E02 ± 5.61E13 1.32E+06 ± 6.44E+05 2.20E02 ± 8.90E03 1.07E+02 ± 2.87E+02 9.10E01 ± 9.21E01 3.29E+00 ± 2.20E+00 7.64E+12 ± 4.18E+12 2.70E08 ± 6.30E09 2.97E08 ± 5.93E09 4.07E08 ± 2.36E08 1.31E02 ± 2.37E02 11/2/0

SaDE + + + + = = + + + + +  +

MVO + = + + = + + + + + + + +

2.87E03 ± 1.35E03 9.89E01 ± 3.77E01 2.24E06 ± 5.43E07 3.15E+06 ± 1.23E+06 8.78E03 ± 3.28E03 2.83E+02 ± 5.55E+02 3.32E+00 ± 2.42E+00 3.16E01 ± 1.22E01 2.74E+15 ± 8.70E+15 4.39E02 ± 1.30E02 4.48E02 ± 1.17E02 2.10E08 ± 1.72E08 2.73E02 ± 1.62E02 11/1/1

4.81E45 ± 1.41E44 6.79E01 ± 4.77E02 0.00E+00 ± 0.00E+00 4.25E38 ± 1.37E37 6.76E03 ± 2.95E03 3.74E+01 ± 2.98E+01 3.57E41 ± 5.59E41 9.63E02 ± 2.40E01 4.25E24 ± 4.34E24 3.30E41 ± 1.15E40 0.00E+00 ± 0.00E+00 7.18E53 ± 3.46E52 6.19E43 ± 1.09E42 1/2/10

 =    =  +     

GWO + = + +  + + + + + + + +

5.55E149 ± 1.70E148 6.67E01 ± 6.10E08 1.04E16 ± 2.82E17 2.50E141 ± 7.98E141 2.43E04 ± 1.54E04 2.63E+01 ± 7.07E01 3.54E143 ± 1.22E142 2.00E35 ± 6.44E35 1.07E81 ± 2.87E81 1.73E144 ± 4.14E144 2.83E01 ± 2.15E01 0.00E+00 ± 0.00E+00 8.46E145 ± 3.58E144 1/3/9

 = =   =     +  

Table 13 Comparison results among different intelligent algorithms for unimodal functions at D ¼ 50. Fun

CMFOA

CLPSO

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 w=t=l

1.43E19 ± 1.19E19 3.70E+00 ± 3.30E+00 1.08E15 ± 1.90E16 1.66E13 ± 1.34E13 8.26E02 ± 2.74E02 1.45E+02 ± 2.88E+02 2.06E20 ± 2.42E20 6.30E03 ± 5.34E03 1.58E09 ± 7.01E10 2.24E18 ± 2.22E18 1.57E18 ± 1.06E18 3.10E26 ± 3.71E26 8.61E19 ± 6.45E19 –

5.64E03 ± 1.67E03 2.11E+01 ± 2.58E+00 4.89E06 ± 1.23E06 3.58E+02 ± 9.70E+01 5.10E02 ± 1.20E02 5.66E+02 ± 1.14E+02 7.88E02 ± 3.29E02 2.90E+01 ± 1.89E+00 1.73E+00 ± 3.04E01 1.03E01 ± 2.62E02 9.91E02 ± 2.43E02 3.04E23 ± 4.18E23 2.10E02 ± 3.75E03 11/2/0

Fun

SGHS

ALO

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 w=t=l

1.26E06 ± 3.60E04 5.86E+00 ± 3.38E+00 0.00E+00 ± 0.00E+00 4.71E18 ± 6.57E18 1.27E01 ± 4.64E02 1.69E+02 ± 3.42E+02 8.13E+03 ± 2.32E+03 5.02E+00 ± 2.38E+00 2.56E02 ± 2.54E02 1.30E01 ± 1.13E01 0.00E+00 ± 0.00E+00 5.89E13 ± 1.48E11 1.04E03 ± 1.27E02 8/2/3

+ =   + = + + + +  + +

7.24E02 ± 1.06E01 1.38E+00 ± 1.38E+00 1.38E11 ± 2.23E12 4.18E+06 ± 1.56E+06 6.62E02 ± 1.75E02 1.63E+02 ± 2.30E+02 1.45E+00 ± 2.01E+00 1.46E+01 ± 3.99E+00 2.70E+43 ± 1.48E+44 3.80E07 ± 1.82E07 4.44E07 ± 3.48E07 3.97E08 ± 2.26E08 1.02E+00 ± 1.36E+00 10/3/0

potential global optimizations at the beginning stage and the local exploitation ability at the last stage to improve the accuracy of the obtained solutions. Hence, by dynamically adjusting the parameter of Entropy En in normal cloud model, a larger search radius is

IASFA + + + + = = + + + + + + +

2.44E+03 ± 1.29E+02 8.43E+05 ± 1.37E+05 9.02E01 ± 8.61E03 3.01E+08 ± 2.78E+08 5.33E+01 ± 9.77E+00 9.24E+07 ± 1.44E+07 1.88E+05 ± 3.83E+04 6.64E+01 ± 1.28E+01 2.47E+51 ± 7.43E+51 4.65E+04 ± 1.91E+03 4.62E+04 ± 2.09E+03 1.41E02 ± 1.11E02 9.34E+03 ± 4.32E+02 13/0/0

SaDE + + + + + + + + + + + + +

MVO + = + + = = + + + + + + +

7.89E02 ± 5.58E02 6.48E+00 ± 6.96E+00 1.71E05 ± 3.35E06 8.92E+06 ± 2.66E+06 2.36E02 ± 7.20E03 2.85E+02 ± 4.81E+02 5.13E+00 ± 5.30E+00 1.85E+00 ± 7.48E01 5.02E+45 ± 2.04E+46 3.38E01 ± 8.05E02 3.48E01 ± 6.89E02 5.44E08 ± 3.96E08 1.79E+00 ± 1.86E+00 10/3/0

3.49E23 ± 9.92E23 4.38E+00 ± 3.44E+00 0.00E+00 ± 0.00E+00 5.41E18 ± 1.50E17 3.28E02 ± 1.11E02 1.08E+02 ± 4.65E+01 3.58E40 ± 4.46E40 2.91E+00 ± 1.71E+00 2.67E13 ± 1.28E12 1.64E21 ± 2.38E21 1.50E21 ± 2.48E21 5.76E29 ± 2.90E28 2.83E22 ± 4.20E22 1/3/9

 =   = =  +     

GWO + = + + = = + + + + + + +

7.70E110 ± 1.89E109 6.67E01 ± 1.18E07 1.30E15 ± 4.21E16 4.79E104 ± 1.41E103 3.69E04 ± 1.73E04 4.64E+01 ± 9.09E01 2.79E143 ± 8.75E143 2.62E23 ± 6.11E23 1.41E61 ± 1.15E61 1.95E107 ± 3.03E107 1.51E+00 ± 4.27E01 0.00E+00 ± 0.00E+00 9.82E108 ± 2.16E107 1/1/11

  =        +  –

assigned in the early stage to improve the global search ability and a smaller search radius is used to fine tune the solution in the last stage. Therefore, the abilities of exploration and exploitation are well balanced.

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L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx Table 14 Comparison results among different intelligent algorithms for multi-modal functions at D ¼ 30. Fun

CMFOA

CLPSO

F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26 F27 w=t=l

1.34E12 ± 5.80E13 8.53E14 ± 4.82E14 1.71E05 ± 4.25E06 1.10E+00 ± 3.63E01 7.81E26 ± 1.11E25 9.43E03 ± 1.11E02 5.68E+00 ± 1.26E+00 3.03E+03 ± 4.15E+03 1.51E+00 ± 3.48E01 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 1.45E+00 ± 3.16E01 1.34E08 ± 1.34E08 1.81E+01 ± 3.44E+01 –

2.26E03 ± 4.03E04 4.53E03 ± 1.23E03 6.61E+00 ± 7.32E01 2.97E+00 ± 3.25E01 5.10E07 ± 2.57E07 2.38E05 ± 1.55E05 5.83E+00 ± 6.29E01 9.18E+03 ± 3.05E+03 3.15E+00 ± 1.79E01 3.28E01 ± 1.59E01 4.29E+00 ± 1.61E+00 4.91E01 ± 5.82E02 1.03E02 ± 1.28E03 6.29E+06 ± 4.38E+06 10/2/2

Fun

SGHS

F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26 F27 w=t=l

6.11E05 ± 4.66E03 4.24E05 ± 4.74E04 8.48E+00 ± 1.67E+00 1.23E+00 ± 4.59E01 8.40E10 ± 7.20E06 7.65E02 ± 6.63E02 5.90E+00 ± 1.43E+00 3.90E+03 ± 4.57E+03 1.85E+00 ± 3.71E01 4.43E02 ± 3.01E01 4.71E07 ± 6.09E07 1.00E+00 ± 1.86E01 7.06E02 ± 1.07E02 1.33E+02 ± 1.53E+02 10/4/0

IASFA + + + + +  = = + + +  + +

ALO + + + + + + = = = + + = + +

1.65E+00 ± 8.37E01 3.93E+00 ± 1.87E+00 1.44E+02 ± 1.46E+01 1.12E+01 ± 7.75E01 5.21E+00 ± 2.15E+00 1.20E02 ± 1.34E02 1.76E+01 ± 2.67E+00 1.54E+02 ± 8.24E+01 8.80E09 ± 4.82E08 7.03E+01 ± 1.73E+01 9.30E+01 ± 3.22E+01 5.13E01 ± 8.60E02 1.68E+01 ± 2.86E+00 6.18E+02 ± 5.70E+01 11/0/3

1.40E+00 ± 1.61E+00 4.88E01 ± 1.19E+00 7.99E+01 ± 3.92E+01 8.04E+00 ± 2.81E+00 8.01E+00 ± 6.46E+00 2.94E01 ± 2.12E01 1.91E+01 ± 1.03E+00 4.40E+03 ± 4.34E+02 5.36E+00 ± 3.43E01 5.09E+01 ± 5.80E+01 7.47E+01 ± 7.10E+01 1.53E+00 ± 1.08E+00 5.09E+00 ± 3.80E+00 5.30E+05 ± 8.37E+05 12/2/0

SaDE + + + + + = + + + + + = + +

MVO + + + + + + +   + +  + +

4.49E01 ± 5.53E01 2.80E+00 ± 1.38E+00 1.02E+02 ± 3.83E+01 1.13E+01 ± 6.04E01 6.10E01 ± 8.09E01 1.80E02 ± 1.09E02 1.83E+01 ± 1.54E+00 3.65E+03 ± 4.60E+03 4.81E+00 ± 3.98E01 9.66E+01 ± 2.46E+01 1.31E+02 ± 3.67E+01 4.97E01 ± 4.90E02 7.56E+00 ± 2.62E+00 7.72E+02 ± 4.23E+01 11/2/1

1.32E01 ± 3.43E01 2.06E08 ± 7.98E08 1.58E01 ± 5.35E01 5.32E+00 ± 3.68E01 1.04E02 ± 4.17E02 4.67E03 ± 1.05E02 7.83E+00 ± 1.22E+00 2.82E+03 ± 1.04E+03 3.62E+00 ± 1.79E01 0.00E+00 ± 0.00E+00 3.38E15 ± 1.52E14 2.83E01 ± 5.31E02 8.24E03 ± 3.14E02 4.83E+01 ± 6.47E+01 9/3/2

+ + + + + = = = +  +  + +

GWO + + + + + + + = = + +  + +

6.99E15 ± 6.49E16 2.25E06 ± 5.47E06 1.03E39 ± 9.07E40 3.55E+00 ± 1.85E+00 2.15E02 ± 1.16E02 0.00E+00 ± 0.00E+00 5.87E01 ± 1.60E+00 4.61E+03 ± 2.40E+02 4.36E+00 ± 4.25E01 0.00E+00 ± 0.00E+00 6.33E01 ± 2.06E+00 1.27E01 ± 4.50E02 2.13E14 ± 0.00E+00 3.95E+02 ± 1.08E+01 6/1/7

 +  + +   = +  +   +

Table 15 Comparison results among different intelligent algorithms for multi-modal functions at D ¼ 50. Fun

CMFOA

CLPSO

F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26 F27 w=t=l

2.34E10 ± 9.80E11 5.97E11 ± 7.06E11 1.72E01 ± 5.34E01 1.99E+00 ± 5.60E01 2.64E21 ± 1.86E21 7.39E03 ± 7.92E03 1.04E+01 ± 1.96E+00 3.71E+04 ± 2.68E+04 2.97E+00 ± 4.23E01 3.31E02 ± 3.04E01 3.33E02 ± 1.83E01 2.25E+00 ± 3.74E01 1.36E06 ± 3.44E07 1.15E+02 ± 2.00E+02 –

1.86E01 ± 3.31E02 1.06E01 ± 3.22E02 4.76E+01 ± 3.95E+00 9.87E+00 ± 9.17E01 1.90E03 ± 4.49E04 6.39E03 ± 1.95E03 1.39E+01 ± 8.99E01 2.04E+05 ± 3.90E+04 6.47E+00 ± 2.52E01 2.28E+01 ± 3.32E+00 3.17E+01 ± 2.80E+00 1.27E+00 ± 1.09E01 4.22E01 ± 3.43E02 4.46E+09 ± 2.08E+09 11/3/0

Fun

SGHS

F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26 F27 w=t=l

6.25E02 ± 3.80E02 1.64E03 ± 2.30E03 2.26E+01 ± 4.18E+00 2.79E+00 ± 6.74E01 1.20E04 ± 5.81E04 2.24E01 ± 1.26E01 1.18E+01 ± 2.26E+00 2.31E+04 ± 4.57E+03 8.48E01 ± 3.71E01 1.44E01 ± 3.01E01 4.71E07 ± 6.61E07 1.00E+00 ± 1.86E01 7.06E02 ± 1.07E02 1.33E+02 ± 1.53E+02 8/4/2

IASFA + + + + + = = + + + + = + +

ALO + + + + + + = =  +  = + =

2.65E+00 ± 4.39E01 9.45E+00 ± 3.01E+00 2.76E+02 ± 3.34E+01 2.00E+01 ± 1.62E+00 8.52E+00 ± 2.16E+00 8.47E03 ± 8.90E03 3.01E+01 ± 5.09E+00 7.04E+03 ± 1.20E+03 7.16E03 ± 3.92E02 1.24E+02 ± 3.04E+01 1.71E+02 ± 5.97E+01 1.00E+00 ± 1.44E01 3.34E+01 ± 5.56E+00 1.98E+03 ± 1.40E+02 9/3/2

1.88E+01 ± 6.25E02 5.18E+01 ± 1.64E03 3.95E+02 ± 2.26E+01 1.82E+01 ± 2.79E+00 4.54E+03 ± 1.20E04 1.27E+01 ± 3.88E01 3.76E+01 ± 1.11E+00 2.01E+07 ± 9.99E+05 1.02E+01 ± 2.96E01 4.39E+02 ± 7.85E+01 2.16E+02 ± 9.88E+01 2.16E+01 ± 6.98E01 5.81E+01 ± 5.80E+00 4.78E+16 ± 9.47E+16 13/1/0

SaDE + + + + + + = + + + + + + +

1.26E+00 ± 5.04E01 1.04E07 ± 2.96E07 2.25E+00 ± 3.53E+00 1.41E+01 ± 5.28E01 2.49E02 ± 5.80E02 9.95E03 ± 2.04E02 2.04E+01 ± 9.89E01 2.15E+04 ± 7.29E+03 7.54E+00 ± 2.14E01 2.05E+00 ± 3.76E+00 2.41E+01 ± 3.67E+00 1.86E01 ± 1.28E01 6.22E01 ± 6.30E01 4.39E+02 ± 5.37E+02 8/5/1

+ + + + + + = = = + +  + +

1.31E14 ± 2.31E15 3.61E05 ± 1.91E04 1.55E29 ± 7.41E30 7.87E+00 ± 2.99E+00 6.22E02 ± 2.82E02 3.56E03 ± 1.95E03 2.35E+00 ± 4.48E+00 2.17E+04 ± 2.01E+02 9.34E+00 ± 5.42E01 0.00E+00 ± 0.00E+00 1.30E+00 ± 2.98E+00 1.60E01 ± 4.98E02 2.84E14 ± 0.00E+00 1.12E+03 ± 1.67E+01 6/2/6

MVO + + + + + = =   + + = + +

1.19E+00 ± 6.42E01 9.74E+00 ± 3.47E+00 2.84E+02 ± 4.95E+01 2.08E+01 ± 6.50E01 2.59E+00 ± 1.22E+00 4.16E02 ± 1.21E02 3.31E+01 ± 1.95E+00 9.54E+04 ± 6.05E+04 9.23E+00 ± 4.16E01 2.27E+02 ± 5.44E+01 2.76E+02 ± 6.55E+01 8.87E01 ± 1.20E01 2.11E+01 ± 4.76E+00 2.38E+03 ± 1.01E+02 10/3/1

+ + + + + = = = = + +  + =

GWO  +  + + =  = +  +   +

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14

L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx

Table 16 Comparison results among different intelligent algorithms for composite functions at D ¼ 30. Fun

CMFOA

CLPSO

IASFA

F28 F29 F30 F31 F32 F33 w=t=l

3.18E13 ± 8.39E14 9.88E+02 ± 6.29E+02 9.70E+01 ± 6.02E+01 1.58E02 ± 1.28E02 3.28E13 ± 6.79E14 1.22E+00 ± 2.97E01 –

3.26E05 ± 1.14E05 9.69E+03 ± 1.62E+03 2.68E+02 ± 7.35E+01 4.70E+03 ± 1.46E01 1.88E01 ± 9.07E02 4.18E+00 ± 5.58E01 6/0/0

Fun

SGHS

F28 F29 F30 F31 F32 F33 w=t=l

1.05E04 ± 9.16E03 3.20E+2 ± 5.29E+02 3.34E+03 ± 4.12E+03 3.71E+03 ± 2.34E+03 1.17E+02 ± 2.56E+02 8.45E+00 ± 2.34E+00 4/2/0

+ + + + + +

4.78E+02 ± 2.78E+02 7.96E+03 ± 2.10E+03 9.04E+07 ± 1.29E+08 3.93E+02 ± 1.64E+02 2.96E+02 ± 1.24E+01 4.69E+01 ± 9.12E+00 6/0/0

ALO + = + + + =

SaDE + + + + + +

MVO

2.59E08 ± 6.28E09 2.07E+01 ± 2.05E+01 2.34E+03 ± 3.32E+03 4.73E+03 ± 1.37E+01 1.17E+02 ± 2.86E+01 9.40E+00 ± 2.47E+00 4/1/1

+  + + + =

  = + + +

0.00E+00 ± 0.00E+00 9.35E+00 ± 9.66E+00 8.70E+01 ± 3.97E+01 4.70E+03 ± 8.84E08 9.95E02 ± 3.04E01 6.13E+00 ± 5.49E01 3/1/2 GWO

4.06E02 ± 1.15E02 4.26E+00 ± 1.39E+00 4.22E+03 ± 5.98E+03 4.74E+03 ± 2.97E+01 8.99E+01 ± 2.51E+01 8.39E+00 ± 2.24E+00 4/1/1

+  + + + =

7.96E+02 ± 5.14E+02 1.13E+04 ± 4.18E+03 3.03E+07 ± 2.28E+07 4.70E+03 ± 1.63E02 9.55E+01 ± 2.97E+01 5.72E+00 ± 2.81E+00 5/1/0

+ + + + + =

Table 17 Comparison results among different intelligent algorithms for composite functions at D ¼ 50. Fun

CMFOA

CLPSO

F28 F29 F30 F31 F32 F33 w=t=l

5.76E13 ± 7.57E14 1.55E+04 ± 4.24E+03 1.43E+03 ± 3.31E+03 1.50E02 ± 8.66E03 3.32E02 ± 1.82E01 2.24E+00 ± 4.56E01 –

2.37E01 ± 5.90E02 4.97E+04 ± 5.14E+03 8.43E+03 ± 2.19E+03 6.24E+03 ± 5.78E+00 2.01E+01 ± 2.44E+00 1.23E+01 ± 9.90E01 4/2/0

Fun

SGHS

ALO

F28 F29 F30 F31 F32 F33 w=t=l

3.31E02 ± 1.33E01 1.14E+04 ± 3.77E+03 3.33E+03 ± 4.51E+03 6.42E+03 ± 5.45E+01 5.75E+02 ± 4.74E+01 3.12E+01 ± 6.25E+00 4/1/1

+ = = + + +

IASFA + = = + + +

SaDE

3.20E+04 ± 2.29E+04 9.10E+03 ± 3.65E+03 4.33E+10 ± 6.61E+09 5.17E+03 ± 4.23E+02 6.89E+02 ± 2.43E+01 6.60E+02 ± 1.95E+02 6/0/0

+ + + + + +

MVO

3.49E07 ± 1.00E07 3.39E+03 ± 1.02E+03 2.32E+03 ± 3.16E+03 6.42E+03 ± 2.45E+01 2.75E+02 ± 4.74E+01 2.62E+01 ± 5.95E+00 4/1/1

+  = + + +

  = + + +

0.00E+00 ± 0.00E+00 1.94E+03 ± 6.81E+02 3.56E+03 ± 2.56E+03 6.20E+03 ± 1.86E04 2.62E+00 ± 2.54E+00 1.58E+01 ± 1.04E+00 3/1/2 GWO

3.72E01 ± 8.94E02 1.66E+02 ± 4.97E+01 3.69E+03 ± 4.65E+03 6.28E+03 ± 2.89E+01 1.78E+02 ± 3.63E+01 1.89E+01 ± 4.20E+00 4/1/1

+  = + + +

4.43E+03 ± 2.54E+03 2.97E+04 ± 7.96E+03 4.45E+08 ± 5.68E+08 6.20E+03 ± 1.46E+00 2.34E+02 ± 4.29E+01 1.83E+01 ± 9.34E+00 5/1/0

+ = + + + +

Table 18 Ranks computed by the Wilcoxon test for different intelligent algorithms on Benchmark functions at D ¼ 30.

CLPSO IASFA SGHS SaDE ALO MVO GWO CMFOA

(1) (2) (3) (4) (5) (6) (7) (8)

(1)

(2)

– 134.0 294.0 399.0 149.0 130.0 326.0 513.0

427.0 – 487.0 532.0 378.5 380.0 451.0 560.0













(3)

(4)

267.0 74.00 – 438.5 171.0 161.0 303.0 508.0

129.0 29.00 122.5 – 47.00 19.00 272.5 287.0





(5)

412.0 182.5 357.0 514.0 – 196.0 417.0 471.0

(6)









431.0 181.0 400.0 542.0 365.0 – 416.0 513.0









(7)

(8)

202.0 110.0 258.0 288.5 144.0 145.0 – 333.0

48.00 1.000 53.00 241.0 90.00 48.00 195.0 –

(8)



Table 19 Ranks computed by the Wilcoxon test for different intelligent algorithms on Benchmark functions at D ¼ 50.

CLPSO (1) IASFA (2) SGHS (3) SaDE (4) ALO (5) MVO (6) GWO (7) CMFOA (8)

(1)

(2)

– 41.00 404.0 451.0 229.0 197.0 418.0 536.0

520.0 – 524.0 543.0 540.0 527.0 521.0 540.0







(3)









157.0 37.00 – 345.0 232.5 154.0 363.0 448.0

(4)



5. Conclusion The FOA is a new approach for finding global optimization based on the food finding behavior of the fruit fly. To improve

110.0 18.00 183.0 – 93.00 48.00 270.0 381.0





(5)

(6)

(7)

332.0 21.00 328.5 468.0 – 182.0 416.0 457.0

364.0 34.00 407.0 513.0 379.0 – 435.0 517.0

143.0 40.00 198.0 258.0 145.0 126.0 – 316.0

















25.00 21.00 113.0 180.0 104.0 44.00 245.0 –



the convergence performance of the FOA, an improved FOA is proposed in this paper by introducing a new solution generating method based on normal cloud generator and a parameter adaptive strategy. The normal cloud model is used to describe the

Please cite this article in press as: L. Wu et al., A cloud model based fruit fly optimization algorithm, Knowl. Based Syst. (2015), http://dx.doi.org/10.1016/j. knosys.2015.09.006

L. Wu et al. / Knowledge-Based Systems xxx (2015) xxx–xxx

randomness and fuzziness of the foraging behavior of fruit fly swarm. By dynamically adjusting the parameter of Entropy En in normal cloud model, a larger search radius is assigned in the early stage to improve the global search ability and a smaller search radius is used to fine tune the solutions in the last stage. Therefore, the abilities of exploration and exploitation are well balanced. To evaluate the effectiveness of the proposed method, the CMFOA was applied to 33 benchmark functions and compared with the basic FOA and several recent variants of FOA such as MFOA, MSFOA and IFFO. The experimental results show that, for Benchmarks with D = 30, the CMFOA obtained 20, 27, 25 and 24 significantly better solutions than the IFFO, MSFOA, MFOA and FOA, and obtained 13, 4, 3 and 7 competitive solutions with the IFFO, MSFOA, MFOA and FOA, respectively. When the dimensions are increased to 50, it obtained 21, 28, 25 and 24 significantly better solutions than the IFFO, MSFOA, MFOA and FOA, and obtained 12, 5, 1 and 4 competitive solutions with the IFFO, MSFOA, MFOA and FOA, respectively. The competitive results indicate that the CMFOA has better global convergence ability compared with the MFOA, IFFO, MSFOA and the original FOA. To further demonstrate its effectiveness, the CMFOA was also compared with seven representative intelligent optimization algorithms such as the CLPSO, SaDE, IASFA, SGHS, ALO, MVO and GWO. The experimental results demonstrate that the CMFOA obtained significantly better solutions than the CLPSO, IASFA, SGHS, ALO and MVO. When it is compared with the SaDE, the CMFOA wins 9 of 14 and 8 of 14 multi-modal functions at D = 30 and D = 50, respectively. But it losses 10 of 13 and 9 of 13 unimodal functions at D = 30 and D = 50, respectively. While compared with the GWO, the CMFOA wins 5 of 6 composite functions both at D = 30 and D = 50 but losses 9 of 13 and 11 of 13 unimodal functions at D = 30 and D = 50, respectively. The results on the unimodal functions testified its local search ability. The global search ability was confirmed by the results on multi-modal functions, and the good balance of global and local search ability was evidenced by the results on composite functions. However, there is still room for further improving the exploitation ability of the CMFOA. For future work, we are planning to further improve the exploitation ability of the CMFOA and apply it to the parameters identification of permanent magnet synchronous motor. Moreover, we are going to extend it to the multi-objective optimization. The application of CMFOA for the large-scale continuous optimization problems [35] is also an interesting future work. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.knosys.2015.09. 006. References [1] J. Kennedy, R.C. Eberhart, Particle swarm optimization, in Proc. IEEE Int. Conf. Neural Network, 4, 1995, pp. 1942–1948. [2] D. Karaboga, B. Akay, A comparative study of artificial bee colony algorithm, Appl. Math. Comput. 214 (2009) 108–132. [3] M. Dorigo, V. Maniezzo, A. Colorni, Ant system: optimization by a colony of cooperating agents, IEEE Trans. Syst. Man Cybern. Part B 26 (1996) 29–41. [4] M. Neshat, G. Sepidnam, M. Sargolzaei, A.N. Toosi, Artificial fish swarm algorithm: a survey of the state-of-the-art, hybridization, combinatorial and indicative applications, Artif. Intell. Rev. 42 (2014) 965–997. [5] D. Dasgupta, Advances in artificial immune systems, IEEE Comput. Intell. Mag. 4 (2006) 40–49. [6] S. Mirjalili, The ant lion optimizer, Adv. Eng. Softw. 83 (2015) 80–98.

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