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Multi-objective immune genetic algorithm solving nonlinear interval-valued programming
Engineering Applications of Artificial Intelligence 67 (2018) 235–245
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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai
Multi-objective immune genetic algorithm solving nonlinear interval-valued programming Zhuhong Zhang *, Xiaoxia Wang, Jiaxuan Lu Department of Big Data Science & Engineering, College of Big Data & Information Engineering, Guizhou University, Guiyang, Guizhou 550025, China
a r t i c l e
i n f o
Keywords: Multi-objective interval-valued programming Immune genetic algorithm Interval analysis Crowding degree model Pareto optimality
a b s t r a c t This work studies one multi-objective immune genetic algorithm with small population to solve a general kind of unconstrained multi-objective interval-valued programming. In this optimization approach, those competitive individuals are discriminated based on interval arithmetic rules and a possibility model; a crowding degree model in interval-valued environments is developed to eliminate redundant individuals; the current population promotes different individuals to evolve towards specific directions by population sorting and immune evolution, while those elitist individuals found accelerate to explore the desired regions through genetic evolution. The theoretical analysis has showed that the computational complexity of the proposed approach depends mainly on the elitist population size. Comparative experiments have illustrated that the approach can take a rational tradeoff between effect and efficiency. It can perform well over the compared approaches as a whole, and has the potential to solving multi-modal and hard multi-objective interval-valued programming problems. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In real-world multiple criteria optimization challenges, a large number of provoking problems, e.g., portfolio investment, collision detection, uncertain control systems, etc., present complex parameter uncertainty. Once their uncertain parameters are bounded, multiobjective interval programming (MIP) can be adopted to depict their attributes, usually including two general kinds of programming models, i.e., multi-objective interval number programming (MINP) and multiobjective interval-valued programming (MIVP). The former is with bounded uncertain parameters, whereas the latter involves in interval parameters. Since MIVP is a striking and provoking topic in the context of uncertain programming, any existing static multi-objective optimization algorithms become difficult in seeking its Pareto optimal solutions. The main challenge includes four points: (i) probabilistic dominance models should be designed to execute individual comparison in interval-valued environments, (ii) the Pareto optimal solutions depend greatly on the upper and lower bounds of uncertain factors, (iii) the strategies of conventional individual comparison, population diversity and population evolution cannot adapt to variable parameters environments, and (iv) although MIVP can be converted into a manyobjective static programming model by means of the midpoints and radiuses of its interval-valued objectives, the computationally expensive
cost is inevitable. Therefore, new intelligent optimization approaches are desired. Despite wide applications, MIVP is a still open topic in the fields of mathematical programming and intelligent optimization. It will become increasingly popular (Jin and Branke, 2005), because lots of realworld optimization problems touch on interval parameters. Usually, the main concern involves three points: (i) since MIVP’s Pareto front consists of a series of hypercubes in the objective space, new or extended individual comparison approaches and also crowding degree models are desired, (ii) some valuable biological inspirations need to be borrowed to construct effective and efficient evolutionary mechanisms, and (iii) by comparison against multi-objective single-valued mathematical programming, MIVP needs more computational resources. Although researchers made great efforts to explore efficient intelligent optimization approaches for such kind of problem, it still remains open to study useful and advanced optimization approaches capable of finding finite Pareto optimal solutions with wide coverage scope and uniform distribution. Based on this consideration,we in the present work develop a new multiobjective interval-valued immune genetic approach (MIIGA) with small population in interval-valued environments in order to solve nonlinear multi-objective interval-valued programming problems, relying upon interval arithmetic rules, simplified immune response mechanisms and additive genetic operators. The main contribution of the present work
* Corresponding author.
E-mail address: [email protected] (Z. Zhang). https://doi.org/10.1016/j.engappai.2017.10.004 Received 16 December 2016; Received in revised form 29 August 2017; Accepted 9 October 2017 Available online 5 November 2017 0952-1976/© 2017 Elsevier Ltd. All rights reserved.
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includes three points: (i) two interval-based computational models are designed to identify the importance and diversity of individual for a given population, which can help us solve the difficulty of individual evaluation in interval-valued environments, (ii) a competitive and hybrid multi-objective optimization algorithm is proposed to solve MIVP, which borrows some biological inspirations from immunology and genetics, and (iii) the current work is helpful for opening the door to investigate hybrid multi-objective interval-valued optimization approaches. It is worth pointing out that, although MIIGA, like other immune genetic algorithms, is designed based on several immune metaphors from the clonal selection principle and a few inspirations presented in the genetics and evolution theory, it differs from any existing multiobjective optimization algorithms because of different problems solving and biological inspirations. Especially, whereas multi-objective immune genetic algorithms have been well studied for static multi-objective optimization problems (Luo et al., 0000), they cannot be directly applied to the case of MIVP, for which the main reason is that, on one hand, each individual should be evaluated based on interval comparison instead of real number, and on the other hand their strategies of selection, individual comparison and population diversity do not suite to multiobjective interval-valued environments. MIIGA is designed specially for MIVP problems rather than static multi-objective problems. It needs not only a computational model suitable for interval-valued environments to decide the importance of each individual in a given population, but also a new crowding degree model to measure the similarity of individual. It should be emphasized that the operators of crossover and mutation from the inspirations of genetics and evolution are only used additively to promote the process of solution search. Additionally, we also notice that immune inspirations are rarely borrowed to design optimizers for MIVP problems. The remaining text in this paper is organized as follows: Section 2 gives a survey of related work concerning evolutionary algorithms and immune optimization for MIP. In Section 3, we describe the multiobjective interval-valued programming problem to be solved. A crowding degree model is developed in Section 4. MIIGA is formulated and designed in detail in Section 5. Further, the computational complexity of MIIGA is analyzed, while three performance criteria are cited to execute algorithm comparison in Section 6. Sections 7 and 8 present the whole experimental study, including the experimental setup, test problems, experimental analysis and an engineering application. We conclude that MIIGA is a competitive optimization tool for MIVP in Section 9.
process industry systems, and gave a sufficient and necessary condition which a feasible solution existed, but did not design a solution search procedure to solve one such multi-objective problem. About model transformation on MIVP, midpoint and radius functions are usually used to change MIVP as a deterministic multi-objective programming problem. For instance, Yokota et al. (1995) made an early contribution to transforming a multi-objective interval-valued integer programming problem into a static single-objective integer programming one. The existing intelligent approaches solving MIVP models can be divided into two broader classes, i.e., model transformation approaches (Li et al., 2010; Cheng et al., 2014; Cheng et al., 2004; Yokota et al., 1995; Li et al., 2011; Bhunia and Samanta, 2014; Jiang et al., 2008; Jiang, 2005; Chen and Chen, 2014) and direct methods (Limbourg and Aponte, 2005; Zhang et al., 2008a; Sun et al., 2013; Zhang et al., 2008b; Zhang et al., 2014; Wang 2014). The former solves MIVP problems by integrating some representative multi-objective evolutionary algorithms (e.g., NSGA-II) with classical programming methods or RBF neural networks. The latter is an interval-valued intelligent optimization approach solving MIVP problems directly, for which the main challenge includes two aspects: (i) discrimination between individuals, namely identification whether an individual is superior to another one, and (ii) crowding degree models. Limbourg and Aponte (2005) based on the basic algorithmic structure of NSGA-II, suggested an extended multiobjective evolutionary algorithm (IP-MOEA) capable of being applied to MIVP problems directly. They designed an individual dominance model by using the left and right endpoints of sub-objective intervalvalued functions, while a hybrid model was developed to measure the similarity of an individual to other individuals in a given population. Additionally, studies on multi-objective evolutionary algorithms, based on possibility degree models have achieved initial success. Gong et al. (2010) and Zhang et al. (2008b) proposed two valuable multi-objective optimization algorithms related to either NSGA-II or particle swarm optimization (PSO), in which some interesting models of probabilistic dominance and crowding degree were designed to identify those valuable individuals and compute their similarity degrees. In order to find the most preferred solutions for MIVP problems, Sun et al. (2013) designed a metric index to emphasize the approximation relationship between candidate solutions and Pareto optimal solutions, and then developed an interactive evolutionary algorithm after incorporating an optimization-cum-decision-making procedure and the idea of preference selection. Zhang et al. (2008a) studied a multi-objective particle swarm optimization algorithm to handle MIVP problems without interval constraints. In one such algorithm, a probable dominance method, used in processing intervals was proposed to identify the qualities of solutions.
2. Related research work The reported studies on MIP concentrate mainly on model transformation and intelligent methodologies. Usually, several conventional approaches, e.g., interval analysis, midpoint and radius methods, weighted coefficient approaches, interval possibility models techniques and so on (Bhunia and Samanta, 2014; Cheng et al., 2014; Cheng et al., 2004; Li et al., 2011; Jiang et al., 2008; Jiang, 2005; Li et al., 2010; Yokota et al., 1995), are adopted to change uncertain objective and constraint functions into deterministic ones. Correspondingly, some achievements on model transformation have appeared in the literature. For example, Li et al. (2010) studied early how to transform MINP into a nested MIVP model by the upper and lower bounds of the objective function and constraints, and subsequently one such model was changed into a static single-objective programming model by means of the penalty function method, possibility models and midpoint and radius functions of its sub-objective interval-valued functions. This is a usual and popular model transformation method. Based on this idea of transformation, Cheng et al. (2014) discussed how MINP models were converted into unconstrained multi-objective programming ones, where the values of the uncertain objective and constraint functions for each candidate were decided by RBF neural networks, instead of Taylor series and interval analysis. Cheng et al. (2004) developed a general MINP model for
3. Problem formulation and basic concepts Let 𝐼𝑅 represent the set of bounded and closed intervals in 𝑅. Each element in 𝐼𝑅 is said to be an interval number. Consider the following general nonlinear multi-objective interval-valued programming problem of form MIVP: min 𝑓 (𝐱, 𝑈 ) = (𝑓1 (𝐱, 𝑈 ), 𝑓2 (𝐱, 𝑈 ), … , 𝑓𝑚 (𝐱, 𝑈 )), 𝐱∈𝐷
with bounded and closed domain 𝐷 in 𝑅𝑝 , and decision vector 𝐱 ∈ 𝐷, where 𝑈 denotes a 𝑞-dimensional interval vector, i.e., the product of 𝑞 interval numbers; 𝑓 (𝐱, 𝑈 ) is the interval-valued vector function. 𝑓𝑖 (𝐱, 𝑈 ) is the ith sub-objective function with 1 ≤ 𝑖 ≤ 𝑚, represented by [𝑓𝑖 (𝐱), 𝑓 𝑖 (𝐱)] by means of interval arithmetic rules. Interval analysis is a valuable tool in the fields of interval programming and dynamic systems. Moore et al. (2009) introduced some basic properties on interval number. They gave the following partial order relation between two interval numbers 𝐴 = [𝐴𝐿 , 𝐴𝑅 ] and 𝐵 = [𝐵 𝐿 , 𝐵 𝑅 ], 𝐴 < 𝐵 ⟺ 𝐴𝑅 < 𝐵 𝐿 . 236
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Fig. 1. The relationship between hypercubes of two candidates.
We here introduce the following weak comparison relation between 𝐴 and 𝐵, 𝐴 ≤ 𝐵 ⟺ 𝐴𝑅 ≤ 𝐵 𝐿 , 𝐴 ≥ 𝐵 ⟺ 𝐵 ≤ 𝐴.
include non-intersection, intersection and inclusion. In the above figure, 𝑋 and 𝑌 denote the hypercubes of interval objective vectors 𝑓 (𝐱, 𝑈 ) and 𝑓 (𝐲, 𝑈 ) for candidates 𝐱 and 𝐲, respectively; A and B represent the vertexes of 𝑋 and 𝑌 at the top-left corner, respectively; 𝑉1 and 𝑉2 stand for the hypervolumes of the corresponding dash hypercubes. Let 𝑉𝑥𝑦 be the hypervolume of the outer hypercube. In this way, 𝑉1 , 𝑉2 and 𝑉𝑥𝑦 can be represented by the endpoints of the interval-valued sub-objectives,
(2)
Generally speaking, it is not true that 𝐴 − 𝐴 = 0, for example when A=[−2,2], we can see that A-A=[−4,4]. Thus, we here prescribe that 𝐴 equals 𝐵 if and only if 𝐴𝐿 = 𝐵 𝐿 and 𝐴𝑅 = 𝐵 𝑅 . This way, Eq. (2) indicates that once 𝐴 ≤ 𝐵 and 𝐵 ≤ 𝐴, 𝐴 and 𝐵 will degenerate to 𝑎, i.e., [𝑎, 𝑎]. It is emphasized that, when 𝐴 ∩ 𝐵 ≠ 𝜙 and 𝐴 ≠ 𝐵, Eq. (2) cannot be adopted to execute comparison between 𝐴 and 𝐵. In such case, the version of possibility degree is an alternative way to compare 𝐴 with 𝐵. We here cite the following possibility degree model (Zhou and Liu, 2008), 𝑑(𝐴, 𝐶) 𝑃 𝑟{𝐴 ≤ 𝐵} = , 𝑑(𝐴, 𝐶) + 𝑑(𝐵, 𝐶)
𝑉1 = 𝑉2 =
|𝑓𝑖 (𝐱) − 𝑓𝑖 (𝐲)|, |𝑓𝑖 (𝐱) − 𝑓𝑖 (𝐲)|,
𝑉𝑥𝑦 =
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(𝑐𝑖 − 𝑑𝑖 ),
𝑖=1
(3) where 𝑐𝑖 = max{𝑓𝑖 (𝐱), 𝑓𝑖 (𝐲)} and 𝑑𝑖 = min{𝑓𝑖 (𝐱), 𝑓𝑖 (𝐲)}. Additionally, the distance between A and B is defined by √ √𝑚 √∑ (8) 𝑑(𝐱, 𝐲) = √ (𝑚(𝑓𝑖 (𝐱, 𝑈 )) − 𝑚(𝑓𝑖 (𝐲, 𝑈 )))2 , 𝑖=1
where 𝑚(𝑓𝑖 (𝐱, 𝑈 )) denotes the midpoint of interval 𝑓𝑖 (𝐱, 𝑈 ). Although Eq. (8) can present the degree of approximation of points A and B, it cannot completely reflect the relation between the two hypercubes. Once we move the positions of 𝑋 and 𝑌 , the values of 𝑉1 , 𝑉2 , 𝑉𝑥𝑦 and 𝑉1 +𝑉2 𝑑(𝐱, 𝐲) will make change correspondingly, but 2𝑉 only does minor 𝑥𝑦 change. This way, the crowding degree of 𝐱 and 𝐲 in the objective space can be defined here by
(5)
𝐱∗
Naturally, is said to be a Pareto optimal solution of the above MIVP if there does not exist 𝐲 in 𝐷 such that 𝐲 ≺ 𝐱∗ . In order to identify whether one individual is superior to another one in a given population 𝐴 with size 𝑁, we introduce the following computational model to compute the rank of individual 𝐱 in 𝐴, 𝑅(𝐱) = 𝑁 − |{𝐲 ∈ 𝐴|𝐲 ≺ 𝐱}|,
𝑖=1 𝑚 ∏
𝑖=1 𝑚 ∏
where 𝐶 = [𝐶 𝐿 , 𝐶 𝑅 ], 𝐶 𝑅 = max Γ and 𝐶 𝐿 = max Γ ⧵ {𝐶 𝑅 } with Γ = {𝐴𝐿 , 𝐴𝑅 , 𝐵 𝐿 , 𝐵 𝑅 }, and meanwhile √ (𝐴𝐿 − 𝐶 𝐿 )2 + (𝐴𝑅 − 𝐶 𝑅 )2 𝑑(𝐴, 𝐶) = . (4) 2 Based on Eq. (3), Zhou and Liu (2008) introduced the version of individual dominance. In other words, 𝐱 dominates 𝐲 (𝐱 ≺ 𝐲) if 𝜎𝑖 (𝐱, 𝐲) ≥ 𝜎𝑖 (𝐲, 𝐱) for any 𝑖 with 1 ≤ 𝑖 ≤ 𝑚, and there exists 𝑖0 such that 𝜎𝑖0 (𝐱, 𝐲) > 𝜎𝑖0 (𝐲, 𝐱), where 𝜎𝑖 (𝐱, 𝐲) = 𝑃 𝑟{𝑓𝑖 (𝐱, 𝑈 ) ≤ 𝑓𝑖 (𝐲, 𝑈 )}.
𝑚 ∏
𝐷(𝐱, 𝐲) =
𝑉1 + 𝑉2 𝑑(𝐱, 𝐲). 2𝑉𝑥𝑦
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𝑉 +𝑉
Again, since 0 ≤ 1𝑉 2 < 1, 𝐷(𝐱, 𝐲) changes within 0 and 𝑑(𝐱, 𝐲). Eq. 𝑥𝑦 (9) hints that if 𝑋 and 𝑌 are close, 𝐷(𝐱, 𝐲) is small. Therefore, in order to reflect the diversity of a given population 𝑃 , the crowding degree of individual 𝐱 in 𝑃 is designed by
(6)
where |Λ| represents the number of elements in set Λ. Eq. (6) indicates that 𝐱 is better if 𝑅(𝐱) is larger, and especially it is a non-dominated individual in 𝐴 if 𝑅(𝐱) = 𝑁. This way, 𝐴 may be divided into two subpopulations. One is composed of non-dominated individuals in 𝐴, the other consists of those dominated ones.
𝐷(𝐲, 𝐱) + 𝐷(𝐱, 𝐳) (10) 2 where 𝐲 and 𝐳 are two closest individuals of 𝐱 in 𝑃 . Eq. (10) illustrates that, if the crowding degrees of individuals in 𝑃 are small, these individuals will have high similarity and thus weak population diversity. 𝐶(𝐱) =
Remark 1. When computing the ranks of all individuals in 𝐴, we first need to calculate the possibility degrees of sub-objective interval numbers for any two individuals through Eq. (3), for which the total of executions is 𝑚𝑁 2 . Second, we also execute 𝑚𝑁 2 comparisons between individuals through non-dominated sorting. Once all the non-dominated individuals in 𝐴 need to be discriminated, the computational complexity is 𝑂(𝑚𝑁 2 ).
Remark 2. In order to compute the crowding degrees of individuals in 𝑃 with size 𝑁, the two closest individuals of each individual in the objective space are first decided, for which we need to execute ( 12 𝑁(𝑁 − 1) + 𝑁𝑙𝑜𝑔𝑁) times. Thus, once Eq. (10) is enforced in 𝑃 , it will cause the complexity of 𝑂(𝑁 2 ). 5. Algorithm formulation and design
4. Crowding degree model Based on the above designs, immune inspirations and also additive genetic operators, we in this section formulates MIIGA, for which the flowchart is given in Fig. 2. One such approach includes three functional modules, i.e., population division, immune evolution and
As we know, the objective interval vectors of individuals 𝐱 and 𝐲, i.e., 𝑓 (𝐱, 𝑈 ) and 𝑓 (𝐲, 𝑈 ), correspond to two hypercubes in the objective space. In general, their possible position relations, given by Fig. 1 237
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high-quality and diverse antibodies. More precisely, after computing the ranks of antibodies in 𝐴 through Eq. (6), population 𝐵 here consists of elements in 𝐴 with the largest rank, namely 𝐵 = {𝐱 ∈ 𝐴|𝑅(𝐱) = max{𝑅(𝐳)|𝐳 ∈ 𝐴}},
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and meanwhile 𝐶 is composed of the elements in 𝐴 but not in 𝐵. (ii) Proliferation & Polynomial(𝐵). Each antibody in 𝐵 proliferates 𝑚𝑐 clones, while changing its genes with mutation rate 𝑝𝑚 through polynomial mutation; subsequently, all the mutated clones constitute clonal population 𝐵 ∗ . (iii) Proliferation & Nonuniform(𝐶). Each antibody in 𝐶 reproduces (𝑚𝑐 − 1) clones. Each clone mutates its genes with mutation rate 𝑝𝑚 through nonuniform mutation; all the mutated clones forms clonal population 𝐶 ∗ . (iv) Selection(𝐻). Such module first computes the ranks of antibodies in 𝐻 through Eq. (6). Second, those non-dominated antibodies constitute a temporary population 𝐷. If |𝐷| > 𝑁, Eq. (10) is used to delete a part of antibodies in 𝐷 with small crowding degrees, and otherwise, some dominated antibodies with large crowding degrees are selected to insert into 𝐷. (v) Recruitment(𝐷). This denotes simply that 𝜇% of elements in 𝐷 are replaced by new antibodies generated randomly. In other words, if 𝐷 only consists of non-dominated antibodies, those elements with small crowding degrees are substituted by some new random antibodies, and otherwise, such new antibodies displace those dominated elements with high similarity. ∗ ). This only keeps those non-dominated (vi) Elitism (𝑀𝑠𝑒𝑡 ∪ 𝑀𝑠𝑒𝑡 memory cells in terms of Eq. (6). Once the number of such cells is beyond 𝑀𝑚𝑎𝑥 , those redundant elements with small crowding degrees vanish from the memory pool by means of Eq. (10).
Fig. 2. The flowchart of MIIGA.
genetic evolution. The current population is divided into two subpopulations; one consists of high-quality individuals, the other is composed of other individuals. The first sub-population not only updates the memory pool, but also participates in evolution through proliferation and polynomial mutation; the second sub-population seeks different kinds of new individuals through nonuniform mutation. The memory pool is mainly used to accelerate its remembers to discover high-quality individuals. Additionally, some random individuals are produced to adjust the diversity of population. In this algorithm statement, we view antigen 𝐴𝑔 as MIVP itself; candidate solutions from the design space 𝐷 are regarded as real-coded antibodies; memory cells are viewed as those best antibodies found until now. Following these descriptions and Fig. 2, MIIGA can be described as given in the form of table at the top of next page. In the above formulation, steps 5.1 to 5.5 are a period of evolution for searching a part of Pareto optimal solutions in MIVP; steps 5.1 and 5.2 are required to determine two sub-populations by means of the ranks of antibodies in 𝐴; step 5.3 produces a new evolving population through immune evolution and population update. Apart from eliminating redundant memory cells, step 5.4 urges those memory cells to discover high-quality antibodies through genetic evolution. More precisely, step 5.4.1 directly eliminates identical or dominated elements, after combining the existing memory cells with those remembers in 𝐵; step 5.4.2 makes the memory set 𝑀𝑠𝑒𝑡 create new antibodies through arithmetic crossover and nonuniform mutation with crossover rate 𝑝𝑐 and mutation rate 𝑝𝑚 ; step 5.4.3 collects those elitist memory cells found. Within a period of iteration, besides the current population is responsible for exploring excellent and diverse antibodies through evolution, a few new random antibodies are inserted to maintain the diversity of population. This cannot only avoid the population to get into local search, but also is helpful for strengthening the ability of global evolution. In order to accelerate the progress of finding diverse Pareto optimal solutions, MIIGA utilizes the additive genetic operators to make those memory cells move towards the desired regions as possible, which can enhance the quality of solution search.
6. Computational complexity and performance criteria 6.1. Computational complexity In this section, MIIGA’s computational complexity is analyzed in order to know which factors influence its performance efficiency. Theorem 1. In the worst case, the computational complexity of MIIGA is 𝑂(𝑁𝑝𝑚𝑐 + (𝑁 + 𝑀𝑚𝑎𝑥 )2 𝑙𝑜𝑔(𝑁 + 𝑀𝑚𝑎𝑥 )).
5.1. Illustration on main modules
Proof. Through MIIGA’s formulation, step 5.1 computes the ranks of antibodies in 𝐴 with the complexity of 𝑂(𝑁 2 ) mentioned in Remark 1. The computational complexity of step 5.3 are decided by Steps 5.3.1 and 5.3.4. To this point, step 5.3.1 produces 𝑁𝑚𝑐 clones by proliferation, among which each clone mutates its genes with at most 𝑝 times. Consequently, the complexity of one such step is 𝑂(𝑁𝑝𝑚𝑐 ) in the worst case. Like step 5.1, step 5.3.4 needs to decide the ranks of antibodies in 𝐻 with the complexity of 𝑂(|𝐻|2 ); once all elements in 𝐻 are nondominated, we will have to compute their crowding degrees with the complexity of 𝑂(|𝐻|2 ) by Remark 2. Since |𝐻| ≤ 2𝑁, the complexity of step 5.3 in the worst case is 𝑂(𝑁𝑝𝑚𝑐 + 𝑁 2 ). On the other hand, the computational complexity of step 5.4 is determined by steps 5.4.1 and 5.4.3. After the memory pool in step 5.4.1 collects those elitist antibodies in 𝐵, those dominated members in 𝑀𝑠𝑒𝑡 will be deleted through Eq. (6), and thus the complexity is at most 𝑂((𝑁 + 𝑀𝑚𝑎𝑥 )2 ) because of |𝐵| ≤ 𝑁. Further, Step 5.4.3 first eliminates those dominated memory cells in ∗ with the same complexity as that in step 5.4.1 because 𝑀𝑠𝑒𝑡 ∪ 𝑀𝑠𝑒𝑡 ∗ |. Afterwards, when the number of non-dominated cells |𝑀𝑠𝑒𝑡 | = |𝑀𝑠𝑒𝑡 is beyond 𝑀, Eq. (10) is used to delete those memory cells with small ∗ |2 𝑙𝑜𝑔|𝑀 ∪ crowding degrees, for which the complexity is 𝑂(|𝑀𝑠𝑒𝑡 ∪𝑀𝑠𝑒𝑡 𝑠𝑒𝑡 ∗ |). Since |𝑀 | ≤ 𝑁 + 𝑀 𝑀𝑠𝑒𝑡 𝑠𝑒𝑡 𝑚𝑎𝑥 in such case, the complexity is at most 𝑂((𝑁 + 𝑀𝑚𝑎𝑥 )2 𝑙𝑜𝑔(𝑁 + 𝑀𝑚𝑎𝑥 )). Summarily, MIIGA’s computational complexity in the worst case is decided by
(i) Division(𝐴). This divides population 𝐴 into two sub-populations of 𝐵 and 𝐶 so that different evolution fashions can be adopted to explore
𝑂 = 𝑂(𝑁 2 ) + 𝑂(𝑁𝑝𝑚𝑐 + 𝑁 2 ) + 𝑂((𝑁 + 𝑀𝑚𝑎𝑥 )2 𝑙𝑜𝑔(𝑁 + 𝑀𝑚𝑎𝑥 )) = 𝑂(𝑁𝑝𝑚𝑐 + (𝑁 + 𝑀𝑚𝑎𝑥 )2 𝑙𝑜𝑔(𝑁 + 𝑀𝑚𝑎𝑥 )). 238
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The proposed approach MIIGA 1. Input: Population size 𝑁, maximal memory size 𝑀𝑚𝑎𝑥 , clonal size 𝑚𝑐 , crossover rate 𝑝𝑐 , mutation rate 𝑝𝑚 and maximal iterative number 𝐺𝑚𝑎𝑥 ; 2. 0 ← 𝑛, 𝑀𝑠𝑒𝑡 ← 𝜙; 3. Generate an initial antibody population 𝐴 of 𝑁 random antibodies; 4. Evaluate antibodies in 𝐴 by interval arithmetic rules; 5. While 𝑛 < 𝐆𝑚𝑎𝑥 do 5.1. Compute the ranks of the antibodies in 𝐴 by Eq. (6); 5.2. Execute population division, (𝐵, 𝐶): =Division(𝐴); % Superior and inferior sub-populations 5.3. Carry out immune evolution on 𝐴: % New high-quality and diverse antibodies 5.3.1. 𝐵 ∗ ∶=Proliferation & Polynomial(𝐵); % Proliferation and Polynomial mutation 5.3.2. 𝐶 ∗ ∶= Proliferation & Nonuniform(𝐶); % Proliferation and nonuniform mutation 5.3.3. 𝐻 ∶= 𝐵 ∪ 𝐵 ∗ ∪ 𝐶 ∗ ; % Population combination 5.3.4. Implement immune selection, 𝐷=Selection(𝐻); % Elitist and diverse antibodies 5.3.5. Create new population, 𝐴=Recruitment(𝐷); % Diversity maintenance 5.4. Enforce genetic evolution on 𝑀𝑠𝑒𝑡 : % Acceleration on solution search 5.4.1. Collect elitism, 𝑀𝑠𝑒𝑡 =Update(𝑀𝑠𝑒𝑡 , 𝐵); % Elitist memory cells ∗ 5.4.2. Perform genetic evolution, 𝑀𝑠𝑒𝑡 =Evolution(𝑀𝑠𝑒𝑡 ); % Crossover and mutation ∗ =Elitism (𝑀 ∗ 5.4.3. Transact memory update,𝑀𝑠𝑒𝑡 𝑠𝑒𝑡 ∪ 𝑀𝑠𝑒𝑡 );% Elitist memory cells 5.5. 𝑛 ← 𝑛 + 1; 6. End while; 7. Output: Pareto optimal solutions in 𝑀𝑠𝑒𝑡 .
are with high quality but also with high similarity and narrow coverage scope; when the second case occurs, X is poor. Thus, when evaluating the quality of X, the following indices should be included. (ii) Coverage width 𝐶𝑊 . This denotes the Hausdorff distance of the two farthest hypercubes over the non-dominated front of solutions in 𝑋, namely (13)
𝐶𝑊 = max max 𝐻(𝑓𝑖 (𝐱, 𝑈 ), 𝑓𝑖 (𝐲, 𝑈 )). 𝐱,𝐲∈𝑋 1≤𝑖≤𝑚
(iii) Solution density (𝑆𝐷). This is used to measure the distribution characteristic of solutions in 𝑋 in the objective space, defined by √ ∑|𝑋| 1 𝑆𝐷 = (𝑠 − 𝑠𝑖 )2 , 𝑖=1 (14) |𝑋| − 1 𝑠𝑖 = min{𝑠(𝑓 (𝐱𝑖 , 𝑈 ), 𝑓 (𝐱𝑗 , 𝑈 ))|𝐱𝑖 , 𝐱𝑗 ∈ 𝑋, 𝑗 ≠ 𝑖}, where 𝑠 denotes the average of 𝑠𝑖 with 1 ≤ 𝑖 ≤ |𝑋|, and 𝑠(𝐱𝑖 , 𝐱𝑗 ) is given by
Fig. 3. Schematic illustration on how to compute the hypervolume of the hypercone enclosed by the origin and the midpoints of non-dominated hypercubes.
𝑠(𝐱𝑖 , 𝐱𝑗 ) = 1+ Through the above inference, we notice that MIIGA’s computational complexity depends on the parameters of 𝑀𝑚𝑎𝑥 , 𝑁, 𝑚𝑐 and 𝑝. This indicates that it is difficult to solve MIVP, due to high computational complexity. Especially, once 𝑁𝑝𝑚𝑐 ≪ 𝑀𝑚𝑎𝑥 , the complexity will be 2 𝑙𝑜𝑔 𝑀 decided by 𝑂(𝑀𝑚𝑎𝑥 𝑚𝑎𝑥 ). This is why we develop an immune genetic algorithm with small population.
𝑔(𝑖, 𝑗, 𝑙) = 2
∑𝑚 𝑙=1
𝑒−𝑔(𝑖,𝑗,𝑙)
1 , √ (𝑓 (𝐱𝑖 ) − 𝑓 (𝐱𝑗 ))2 + (𝑓 𝑙 (𝐱𝑖 ) − 𝑓 𝑙 (𝐱𝑗 ))2 𝑙
𝑙
𝑚 ∑ max{0, min{𝑤𝑙 (𝑖, 𝑗), 𝑤𝑙 (𝑗, 𝑖), 𝑤𝑙 (𝑖, 𝑖), 𝑤𝑙 (𝑗, 𝑗)}} , 𝑤𝑙 (𝑖, 𝑖) + 𝑤𝑙 (𝑗, 𝑗) 𝑙=1
(15)
𝑤𝑙 (𝑖, 𝑗) = 𝑓 𝑙 (𝑥𝑖 ) − 𝑓 (𝐱𝑗 ), 𝑤𝑙 (𝑗, 𝑖) = 𝑓 𝑙 (𝑥𝑗 ) − 𝑓 (𝐱𝑖 ). 𝑙
𝑙
Eq. (14) indicates that if 𝑆𝐷 is small, the solutions, presented in 𝑋 are with uniform distribution, and conversely such solutions are with poor distribution. (iv) Uncertain degree (𝑈 𝐷). This measures whether the hypercones of all elements in 𝑋 in the interval-valued objective space are small. It is simply defined as the total of hypervolumes of the elements in the objective space. If 𝑈 𝐷 is small, such elements are said to be with low uncertainty. Through the designs of the above indices, if algorithm 𝐴 can acquire smaller values on 𝐻𝑀, 𝑆𝐷 and 𝑈 𝐷 as well as a larger one on 𝐶𝑊 than algorithm 𝐵, the former is better.
6.2. Performance criteria Since the Pareto front of a given MIVP problem consists of a series of hypercubes, the reported performance criteria, used in comparing static multi-objective intelligent approaches is not suitable for multi-objective interval-valued intelligent approaches. In order to examine MIIGA’s performance, we here improve and cite three reported performance criteria (Gong et al., 2010; Wang, 2014) to measure the quality of a given finite solution set 𝑋. (i) Hypervolume measure (HM ). 𝐻𝑀, illustrated by Fig. 3 above represents the hypervolume of the hypercones enclosed by the thick lines and the origin. Since the Pareto front of a given MIVP problem is composed of a series of hypercubes rather than a series of points, in the case where HM is small, it is possible that two cases will take place, namely (a) all solutions in X are close to the Pareto front in the objective space, and (b) X includes fewer points, while these points are similar and far from the Pareto front. The first case shows that all the solutions in X
7. Experimental study Our experiments are executed on a computer with CPU/3.00 GHz and RBM/2.00 GB by means of Visual C++ platform. As associated to the following six multi-objective interval-valued test problems acquired through modifying the reported static benchmark problems, the proposed approach, MIIGA as in Section 5, is compared against 239
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(a) KURI .
(b) POLI .
(c) ZDT1I .
(d) ZDT2I .
(e) ZDT3I .
(f) ZDT4I .
Fig. 4. Comparison of box plot on 𝐻𝑀.
three competitive intelligent optimizers found, namely two multiobjective evolutionary algorithms (G-MOEA (Gong et al., 2010) and IP-MOEA (Limbourg and Aponte, 2005)) and one multi-objective particle swarm optimization algorithm so-called FD-MOPSO (Zhang et al., 2014). These compared approaches are specially designed to solve MIVP problems, based on the conventional multi-objective intelligent optimization approaches. Additionally, the existing static multi-objective intelligent approaches are usually only suitable for static multi-objective problems. In order to examine whether one such type of algorithm can solve MIVP problems, we take two static multi-objective approaches, i.e., multi-objective genetic algorithm NSGA-II (Deb et al., 2002) and multi-objective immune genetic algorithm MIGA (He et al., 2010) for example to handle the static multi-objective problems in which their sub-objectives are the midpoint functions of MIVP’s interval-valued sub-objectives. In order to ensure the fairness of comparison, all the approaches terminate their solution search procedures only when their respective iterative number is 500, while each approach executes 100 single runs on each test example. The settings of parameters for the compared approaches are the same as those presented in the corresponding literatures, for example, IP-MOEA and G-MOEA takes their population sizes as 50, and FDMOPSO’s population size is defined as 10. Through the above analysis on computational complexity as in Section 6, 𝑁, 𝑚𝑐 and 𝑀𝑚𝑎𝑥 , included in MIIGA are three crucial parameters to influence the efficiency of solution search. In order that MIIGA takes a trade-off between efficiency and effect, 𝑁 and 𝑚𝑐 are required to take small values as possible, i.e., 𝑁 ∈ [5, 10] and 𝑚𝑐 ∈ [2, 10]. In addition, 𝑀𝑚𝑎𝑥 is the size of the outer archive set for storing those elitist solutions acquired, it takes a somewhat large value, i.e., 𝑀𝑚𝑎𝑥 ∈ [50, 80]. After manually
experimental tuning, we take 𝑁 = 10, 𝑚 = 50, 𝑚𝑐 = 2, 𝑝𝑐 = 0.45 and 𝑝𝑚 = 0.1. 7.1. Multi-objective interval-valued test problems Limbourg and Aponte (2005) proposed a valuable approach that transformed static multi-objective benchmark problems into multiobjective interval-valued programming ones, in which a disturbed vector-valued function was designed to add to static objective functions. More precisely, let 𝛿(𝐱) be a two-dimensional vector-valued function, 𝛿(𝐱) = (cos(10𝜋
𝑛 ∑ 𝑖=1
𝑥𝑖 )∕2, cos(20𝜋
𝑛 ∑
𝑥𝑖 )∕2)𝑇 .
(16)
𝑖=1
In this way, the following static multi-objective problem min 𝑓 (𝐱) = (𝑓1 (𝐱), 𝑓2 (𝐱), … , 𝑓𝑚 (𝐱)) 𝐱∈𝐷
can be changed as the multi-objective interval-valued problem MIVP in Section 3, where 𝑓𝑖 (𝐱, 𝛿) = [𝑓 (𝐱, 𝛿), 𝑓 𝑖 (𝐱, 𝛿)], with 𝑓 (𝐱, 𝛿) = 𝑖
𝑖
min{𝑓𝑖 (𝐱), 𝑓𝑖 (𝐱) + 𝛿1 (𝐱)} and 𝑓 𝑖 (𝐱, 𝛿) = max{𝑓𝑖 (𝐱), 𝑓𝑖 (𝐱) + 𝛿2 (𝐱)}. 𝛿1 (𝐱) and 𝛿2 (𝐱), which change over an interval [−0.5, 0.5], are the first and second elements of 𝛿(𝐱), respectively. Based on the above problem transformation, we here transform in turn six static multi-objective benchmark problems (KUR, POL, and ZDT1 to ZDT4) (Deb et al., 2002) into multi-objective programming problems (KUR𝐼 , POL𝐼 and ZDT1𝐼 to ZDT4𝐼 ). In order to examine the ability of dealing with interval-valued programming problems for the above approaches, the dimensions of the test problems ZDT1 to ZDT3 are set as 30, but the dimension of ZDT4 is considered as 10. Among these static test problems, the two multimodal optimization problems POL and KUR with dimensions 2 and 240
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(a) KURI .
(b) POLI .
(c) ZDT1I .
(d) ZDT2I .
(e) ZDT3I .
(f) ZDT4I .
Fig. 5. Comparison of box plot on 𝐶𝑊 .
Figs. 4 and 5 show that the above six approaches have different solution qualities and search stability in terms of their quartiles. We notice that NSGA-II and MIGA as two static multi-objective intelligent approaches result in poor search performances by comparison with the other four approaches designed specially for MIVP problems. Whereas NSGA-II can get smaller means on 𝐻𝑀 for all the test examples, so are its means on 𝐶𝑀, which means that one such approach can only find some solutions with high similarity and narrow coverage scope and meanwhile it gets easily into local search. MIGA can obtain some solutions with relatively wide coverage scope for some test problems by Figs. 5 and 8, and thus is with strong population diversity, whereas its solution qualities are both poor and instable by Fig. 4. The main reason involves two points: (a) it can only find some poor solutions for KURI and instable solution qualities for the other test problems because of wide regions between the upper and lower quartiles, and (b) it produces many singular points outside its upper and lower quartiles inFig. 4, which hints that its solution quality acquired for each test problem cannot keep stable. IP-MOEA and G-MOEA perform poor over FD-MOPSO and MIIGA, for which the main reason includes three points: (a) IP-MOEA results in the serious phenomenon of fluctuation of solution search for the test examples except for ZDT4I , while some singular points in its box plots are found. Thereby, by comparison against G-MOEA its solution qualities for the test problems are poor but superior to those acquired by NSGA-II and MIGA, and (b) although G-MOEA can acquire satisfactory solution qualities for POLI , ZDT2I and ZDT3I , it produces some singular points for some test problems, and meanwhile the hypercubes of its solutions for the test problems are with relatively high uncertainty which can be known below.
3 respectively include multiple noncontinuous Pareto front segments. In addition, the test problems ZDT1 to ZDT4 are a set of typical and difficult benchmark problems with different attributes. ZDT1 and ZDT2 have similar characteristics, namely their Pareto fronts are continuous, smooth, and convex for ZDT1 but concave for ZDT2. The Pareto front of ZDT3 consists of 5 disconnected curve segments. Note that the reason why ZDT4 here takes dimension 10 instead of 30, is that it includes 219 local Pareto fronts but only one global Pareto front and hence it is extremely difficult to search the global Pareto front. Since all these static problems are hard and 𝛿1 (𝐱) and 𝛿2 (𝐱) are two uncertain factors, their corresponding multi-objective interval-valued programming problems can be selected to examine the performance characteristics of the above algorithms. 7.2. Experimental results and analysis After respectively executing 100 runs on each interval-valued problem above, the algorithms acquire their respective solution sets. Correspondingly, their statistical characteristics, based on the performance criteria in Section 7 are presented by box-plots through Figs. 4–7. Additionally, we only take KUR𝐼 for example to present the solution distribution in the rectangle objective space, due to the limit of page space. The non-dominated fronts formed by the rectangles of the objective interval vectors, gotten by them after respectively a single run are drawn in Fig. 8. The experimental results, presented by box plots in Figs. 4–8 expose some significant differences between the above four approaches. We can draw the following conclusions: 241
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(a) KURI .
(b) POLI .
(c) ZDT1I .
(d) ZDT2I .
(e) ZDT3I .
(f) ZDT4I .
Fig. 6. Comparison of box plot on 𝑆𝐷. Table 1 Comparison on average runtime (second) spent by each approach. Below A, B, C, D, E and F denote orderly by IP-MOEA, G-MOEA,FD-MOPSO,NSGA-II,MIGA and MIIGA.
By comparing MIIGA with FD-MOPSO, the former is superior to the latter for KURI , POLI and ZDT4I in terms of Figs. 4 and 5. Additionally, whereas MIIGA gains some relatively larger means on 𝐻𝑀 than FDMOPSO for ZDT1I , ZDT2I and ZDT3I by Fig. 4, by Fig. 5 it acquires wider coverage scopes and low uncertainties of solutions (see Fig. 7) for such problems. Therefore, MIIGA is globally superior to FD-MOPSO with the aspect of solution quality (also see Fig. 8). Fig. 6 illustrates that the non-dominated solutions, acquired by MIIGA per run for east test problem are with satisfactory and stable solution distribution; IP-MOEA, G-MOEA and FD-MOPSO are secondary. NSGA-II and MIGA can only find some solutions with poor distribution and distributional instability. Fig. 7 hints that the non-dominated solutions, gotten by NSGA-II and MIGA are with high uncertainty, namely the widths of hypercubes of the solutions are large, and thereby such two approaches as static optimization ones cannot gain satisfactory effects for MIVP problems. Additionally, the non-dominated solutions, found by IP-MOEA and G-MOEA for the test problems have relatively large hypervolumes, and consequently cause relatively high uncertainties but superior to those gotten by NSGA-II and MIGA. Further, the solutions, obtained by MIIGA for ZDT1𝐼 and ZDT3𝐼 are with lower uncertainty than those acquired by FD-MOPSO, while FD-MOPSO can gain some solutions with somewhat lower uncertainty for KUR𝐼 , POL𝐼 , ZDT2𝐼 and ZDT4𝐼 . Thereby, FD-MOPSO is somewhat superior to MIIGA with regard to solution uncertainty, but it produce some large hypercubes for some runs and test examples, e.g. KUR𝐼 (see Fig. 8). With respect to performance efficiency, Table 1 illustrates that for each test problem, FD-MOPSO spends the least runtime, while Figs. 4–8
A B C D E F
KUR𝐼
POL𝐼
ZDT1𝐼
ZDT2𝐼
ZDT3𝐼
ZDT4𝐼
0.48 6.12 0.14 7.28 11.7 10.68
0.42 6.18 0.14 7.8 16 13.46
0.5 5.92 0.18 6.62 7.26 3.1
0.5 5.9 0.12 6.34 7.42 2.22
0.5 5.96 0.2 5.98 7.32 3.38
0.48 6.06 0.12 6.88 9.56 2.46
show that it performs well over IP-MOEA, G-MOEA, NSGA-II and MIGA but poor over MIIGA. Thereby, FD-MOPSO is a competitive optimizer. Whereas IP-MOEA only needs less runtime to solve each test problem by comparison with G-MOEA, NSGA-II, MIGA and MIIGA, its solution search effect is poor. MIIGA can spends less runtime to handle ZDT1I , ZDT2I , ZDT3I and ZDT4I by comparison against G-MOEA, NSGA-II and MIGA. When solving KURI and POLI , it is only more efficient than MIGA, for which the main reason consists in that such two examples are lowdimensional and easy to solve while its memory pool needs to eliminate many old memory cells within each period of evolution. Summarily, when solving the above higher-dimensional hard interval-valued problems, the six approaches present significant differences. MIIGA performs globally well over the compared approaches according to the performance criteria in Section 6.2, for which the main reason consists in that it inherits the merits of immune optimization and genetic algorithms, and meanwhile the proposed crowding degree model can produce the strong diversity of population and strengthen the ability of individual selection in the process of solution search. In 242
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Engineering Applications of Artificial Intelligence 67 (2018) 235–245
(a) KURI .
(b) POLI .
(c) ZDT1I .
(d) ZDT2I .
(e) ZDT3I .
(f) ZDT4I .
Fig. 7. Comparison of box plot on 𝑈 𝐼.
Fig. 8. Comparison of non-dominated fronts acquired for the six approaches with a run for KURI .
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(a) 𝐻𝑀.
(b) 𝐶𝑊 .
(c) 𝑆𝐷.
Fig. 9. Comparison of box plot acquired by the six approaches for EED.
̄ 𝑃̄𝐿 ; Λ(𝑃𝐺 , 𝛿) = where Λ(𝑃𝐺 , 𝛿) = [Λ(𝑃𝐺 , 𝛿), Λ(𝑃𝐺 , 𝛿)] with Λ = 𝐹̄ , 𝐸, 𝑚𝑖𝑛{Λ(𝑃𝐺 ), Λ(𝑃𝐺 ) + 𝛿1 (𝑃𝐺 )}, and Λ(𝑃𝐺 , 𝛿) = 𝑚𝑎𝑥{Λ(𝑃𝐺 ), Λ(𝑃𝐺 ) + 𝛿2 (𝑃𝐺 )}. 𝛿1 (𝑃𝐺 ) and 𝛿2 (𝑃𝐺 ) are mentioned in Section 7.1. This model is solved based on the standard IEEE 30-bus 6-generator test system. All the parameter settings but 𝑁 = 6, 𝑃𝐷 = 2.834, 𝑃𝐺𝑚𝑖𝑛 = 0.05, 𝑃𝐺𝑚𝑎𝑥 = 1.5 with 𝑖 𝑖 1 ≤ 𝑖 ≤ 6 can be found in Wu (2007). The above fix approaches are then tested on the above interval programming model. Their respective maximal iterative number is 500 in a run, and meanwhile each algorithm executes 100 single runs. Like the above experiments, their box plots on indices 𝐻𝑀, 𝐶𝑊 and 𝑆𝐷 are given in Fig. 9. The box plots in Figs. 9(a) and (b) hints that MIIGA can gain the best solution quality among the six approaches while being capable of performing stable solution search; FD-MOPSO is secondary. Although G-MOEA produces several singular points, it behaves stable search performance. Nevertheless, NSGA-II and MIGA can only present the worst search performances, and thus are further confirmed to be not suitable for MIVP problems. Hence, only when being made major improvements, they are desired for MIVP. Additionally, IP-MOEA is also difficult for one such problem, as it presents the phenomenon of instable solution search. Fig. 9(c) illustrates that G-MOEA, FD-MOPSO, MIGA and MIIGA can find some solutions with stable and satisfactory distribution, whereas IP-MOEA and NSGA-II are opposite.
addition, FD-MOPSO is a competitive optimizer for MIVP. NSGA-II and MIIGA as two static optimization approaches cannot effectively solve MIVP problems even if they are competitive for static multi-objective optimization problems. 8. Engineering application Generally, the environmental/economic power dispatch problem needs to optimize simultaneously two competitive objective functions, i.e., fuel cost 𝐹 (𝑃𝐺 ) and emission 𝐹 (𝑃𝐺 ), while satisfying several constraint limits. Such two functions are defined usually by 𝐹 (𝑃𝐺 ) = 𝐸(𝑃𝐺 ) =
𝑁 ∑ (𝑎𝑖 + 𝑏𝑖 𝑃𝐺𝑖 + 𝑐𝑖 𝑃𝐺2 ), 𝑖=1 𝑁 ∑
𝑖
(17) −2
{10
𝑖=1
(𝛼𝑖 + 𝛽𝑖 𝑃𝐺𝑖 + 𝛾𝑖 𝑃𝐺2 ) + 𝜉𝑖 𝑖
exp(𝜆𝑖 𝑃𝐺𝑖 )},
where 𝑃𝐺 = (𝑃𝐺1 , 𝑃𝐺2 , … , 𝑃𝐺𝑁 ), 𝑃𝐺𝑚𝑖𝑛 ≤ 𝑃𝐺𝑖 ≤ 𝑃𝐺𝑚𝑎𝑥 , 1 ≤ 𝑖 ≤ 𝑁; 𝑁 𝑖 𝑖 is the number of generators; 𝑎𝑖 , 𝑏𝑖 and 𝑐𝑖 are the cost coefficients for the ith generator, and meanwhile 𝛼𝑖 , 𝛽𝑖 , 𝛾𝑖 , 𝜉𝑖 and 𝜆𝑖 are the emission coefficients; 𝑃𝐺𝑖 is the real power output of the ith generator. However, another reported and important function 𝑃𝐿 , line loss, is not taken as an objective function in general, given by 𝑃𝐿 (𝑃𝐺 ) =
𝑁 𝑁 ∑ ∑ 𝑗=1 𝑖=1
𝑃𝐺𝑖 𝐵𝑖𝑗 𝑃𝐺𝑗 +
𝑀 ∑
𝐵0𝑖 𝑃𝐺𝑖 + 𝐵00 ,
(18)
9. Conclusions and further work
𝑖=1
where 𝐵𝑖𝑗 , 𝐵0𝑖 and 𝐵00 are the transmission network power loss coefficients for the ith generator. This way, Wu (2007) proposed a threeobjective Environmental/Economic power dispatch (EED) programming model given by
Based on the fact that multi-objective interval-valued programming is a hard topic with comprehensive application background, this work aims at studying an immune genetic optimization approach. We first develop a model to identify the importance of individual in a given population, relying upon the reported concepts of possibility degree and interval dominance. Second, a crowding degree model is designed to determine the degree of similarity for an individual to other individuals, used in picking up diverse individuals in the population. Finally, an immune genetic evolutionary mechanism is designed to seek those high-quality solutions, in which an immune evolutionary mechanism is developed to make the current population explore those diverse and elitist individuals through proliferation, immune selection, and different kinds of mutation operations. The genetic evolutionary mechanism as an additional operator accelerates those remaining memory cells to explore the desired solutions in the design space. The proposed method, with the merits of small population and strong evolution, only includes three adjustable parameters 𝑚𝑐 , 𝑝𝑐 and 𝑝𝑚 . The theoretical analysis has showed that MIIGA’s computational cost is decided mainly by 𝑀𝑚𝑎𝑥 . Comparative experiments have demonstrated that the approach is useful in
min 𝐻(𝑃𝐺 ) = (𝐹 (𝑃𝐺 ), 𝐸(𝑃𝐺 ), 𝑃𝐿 (𝑃𝐺 )), 𝑥∈𝐷
s. t.
𝑁 ∑
𝑃𝐺𝑖 − 𝑃𝐷 − 𝑃𝐿 (𝑃𝐺 ) = 0
𝑖=1
where the equality means that the total power generation must include the real total power demand 𝑃𝐷 and power loss 𝑃𝐿 in transmission lines. When handling the equality constraint, each sub-objective function is ∑ attached here by a penalty function, i.e., | 𝑁 𝑖=1 𝑃𝐺𝑖 − 𝑃𝐷 − 𝑃𝐿 (𝑃𝐺 )|. Like the model transformation fashion in Section 7.1, by means of Eq. (16) we transform the above static three-objective optimization problem into the following three-objective interval-valued problem, ̄ 𝐺 , 𝛿), 𝑃̄𝐿 (𝑃𝐺 , 𝛿)), min 𝐻(𝑃𝐺 , 𝛿) = (𝐹̄ (𝑃𝐺 , 𝛿), 𝐸(𝑃 𝑥∈𝐷
s. t.
𝑁 ∑
𝑃𝐺𝑖 − 𝑃𝐷 − 𝑃𝐿 (𝑃𝐺 ) = 0,
𝑖=1
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practice and performs well over the compared approaches while NSGAII and MIGA as static multi-objective approaches cannot effectively deal with MIVP. In this work, we, however, only make some preliminary investigations on how to explore immune genetic optimization to solve multi-objective interval-valued programming problems. Whereas MIIGA exhibits some excellent properties, some improvements need to be made. For example, its structures will be optimized further in the precondition of improving the solution quality, and also its engineering applications will be further investigated. Additionally, we claim that MIIGA is a hybrid optimization approach which skillfully merges immune and genetic operators into its evolutionary mechanism. This is the first hybrid intelligent optimization approach for MIVP problems. Meanwhile, it is available to investigate hybrid multi-objective intervalvalued intelligent methods by merging multiple thoughts of different optimizers and designing some new operators in multi-objective intervalvalued environments, which is an open topic.
He, G., Gao, J., Hu, L., 2010. An improved immune genetic algorithm for multiobjective optimization. In: International Conference on Advances in Swarm Intelligence. Springer–Verlag, pp. 643–650. Jiang, C., Han, X., Liu, G.R., et al., 2008. A nonlinear interval number programming method for uncertain optimization problems. Eur. J. Oper. Res. 188, 1–13. Jiang, Z., 2005. Optimization of uncertain systems with interval parameters and its application in Gasoline Blending (Ph.D. dissertation), Zhejiang University, China. Jin, Y.C., Branke, J., 2005. Evolutionary optimization in uncertain environments: A survey. IEEE Trans Evol Comput. 9 (3), 303–317. Li, F.Y., Li, G.Y., Zheng, G., 2010. Uncertain multi-objective optimization method based on interval. Chinese Journal of Solid Mechanics 31 (1), 86–93. Li, X.L., Jiang, C., et al., 2011. An uncertainty multi-objective optimization based on interval analysis and its application. Chinese Mechanical Engineering 22 (9), 1100– 1106. Limbourg, P., Aponte, D.E.S., 2005. An optimization algorithm for imprecise multiobjective problem functions. In: 2005 IEEE Congress on Evolutionary Computation, vol. 1, pp. 459–466. Luo, G.F., Wen, X.Y., Li, H., 0000. An effective multi-objective genetic algorithm based on immune principle and external archive for multi-objective integrated process planning and scheduling. Int. J. Adv. Manuf. Technol, http://dx.doi.org/10.1007/s00170-0170020-z. Moore, R.E., Kearfott, R.B., Cloud, M.J., 2009. Introduction to interval analysis. Society for Industrial & Applied Mathematics, US. Sun, J., Gong, D.W., Ji, X.F., 2013. Interactive evolutionary algorithms for interval multiobjective optimization problems based on preference direction. Control and Decision 28 (4), 542–546. Wang, C., 2014. Research on the knowledge-inducing interval multi-objective evolutionary optimization method (Ph.D. dessertation), China University of Mining and Technology. Wu, 2007. Research on multi-objective dynamic differential evolution and its application (Ph.D. disseration), Hunan university, Changsha, China. Yokota, T., Gen, M., Li, Y., 1995. A genetic algorithm for interval nonlinear integer programming problem. In: 18th International Conference on Computers and Industrial Engineering, Great Britain, pp. 913–917. Zhang, E.Z., Wu, Y.F., Chen, Q.W., 2014. Particle swarm optimization algorithms for interval multi-objective optimization problems. Control and Decision 29 (12), 2171– 2176. Zhang, Y., Gong, D.W., Hao, G.S., et al., 2008a. Particle swarm optimization for multiobjective systems with interval parameters. ACTA Automatica Sinica 34 (8), 921– 928. Zhang, Y., Gong, D.W., et al., 2008b. Particle swarm optimization for multi-objective systems with interval parameters. Acta Automatica Sinica 34 (8), 921–928. Zhou, G.M., Liu, S.R., 2008. A new ranking approach for interval numbers in uncertain multiple-attribute decision making problems. Statistics and Decision 19, 154–157.
Acknowledgment The authors would like to thank the cordial editors and reviewers for their contributions to this work. This research is supported by National Natural Science Foundation NSFC (61563009). References Bhunia, A.K., Samanta, S.S., 2014. A study of interval metric and its application in multiobjective optimization with interval objectives. Computers & Industrial Engineering 74, 169–178. Chen, Z.W., Chen, L., 2014. Improved NSGA-II for constrained multi-objective optimization problems with interval numbers. Journal of Chinese Computer Systems 35 (11), 2502–2506. Cheng, J., Duan, G.F., et al., 2014. Interval multi-objective optimization of structures based on radial basis function, interval analysis, and NSGA-II. Journal of Zhejiang University-Science 15 (10), 774–788. Cheng, Z., Dai, L., Sun, Y., 2004. Feasibility analysis for optimization of uncertain systems with interval parameters. Acta Automatica Sinica 30, 455–459. Deb, K., Pratap, A., et al., 2002. A fast and elitist multiobjective genetic algorithm: NSGAII. Evolutionary Computation, IEEE Transactions on 6 (2), 182–197. Gong, D.W., Qin, N.N., Sun, X.Y., 2010. Evolutionary algorithms for multi-objective optimization problems with interval parameters. In: 2010 IEEE Fifth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA), pp. 411– 420.
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Contents lists available at ScienceDirect
Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai
Multi-objective immune genetic algorithm solving nonlinear interval-valued programming Zhuhong Zhang *, Xiaoxia Wang, Jiaxuan Lu Department of Big Data Science & Engineering, College of Big Data & Information Engineering, Guizhou University, Guiyang, Guizhou 550025, China
a r t i c l e
i n f o
Keywords: Multi-objective interval-valued programming Immune genetic algorithm Interval analysis Crowding degree model Pareto optimality
a b s t r a c t This work studies one multi-objective immune genetic algorithm with small population to solve a general kind of unconstrained multi-objective interval-valued programming. In this optimization approach, those competitive individuals are discriminated based on interval arithmetic rules and a possibility model; a crowding degree model in interval-valued environments is developed to eliminate redundant individuals; the current population promotes different individuals to evolve towards specific directions by population sorting and immune evolution, while those elitist individuals found accelerate to explore the desired regions through genetic evolution. The theoretical analysis has showed that the computational complexity of the proposed approach depends mainly on the elitist population size. Comparative experiments have illustrated that the approach can take a rational tradeoff between effect and efficiency. It can perform well over the compared approaches as a whole, and has the potential to solving multi-modal and hard multi-objective interval-valued programming problems. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In real-world multiple criteria optimization challenges, a large number of provoking problems, e.g., portfolio investment, collision detection, uncertain control systems, etc., present complex parameter uncertainty. Once their uncertain parameters are bounded, multiobjective interval programming (MIP) can be adopted to depict their attributes, usually including two general kinds of programming models, i.e., multi-objective interval number programming (MINP) and multiobjective interval-valued programming (MIVP). The former is with bounded uncertain parameters, whereas the latter involves in interval parameters. Since MIVP is a striking and provoking topic in the context of uncertain programming, any existing static multi-objective optimization algorithms become difficult in seeking its Pareto optimal solutions. The main challenge includes four points: (i) probabilistic dominance models should be designed to execute individual comparison in interval-valued environments, (ii) the Pareto optimal solutions depend greatly on the upper and lower bounds of uncertain factors, (iii) the strategies of conventional individual comparison, population diversity and population evolution cannot adapt to variable parameters environments, and (iv) although MIVP can be converted into a manyobjective static programming model by means of the midpoints and radiuses of its interval-valued objectives, the computationally expensive
cost is inevitable. Therefore, new intelligent optimization approaches are desired. Despite wide applications, MIVP is a still open topic in the fields of mathematical programming and intelligent optimization. It will become increasingly popular (Jin and Branke, 2005), because lots of realworld optimization problems touch on interval parameters. Usually, the main concern involves three points: (i) since MIVP’s Pareto front consists of a series of hypercubes in the objective space, new or extended individual comparison approaches and also crowding degree models are desired, (ii) some valuable biological inspirations need to be borrowed to construct effective and efficient evolutionary mechanisms, and (iii) by comparison against multi-objective single-valued mathematical programming, MIVP needs more computational resources. Although researchers made great efforts to explore efficient intelligent optimization approaches for such kind of problem, it still remains open to study useful and advanced optimization approaches capable of finding finite Pareto optimal solutions with wide coverage scope and uniform distribution. Based on this consideration,we in the present work develop a new multiobjective interval-valued immune genetic approach (MIIGA) with small population in interval-valued environments in order to solve nonlinear multi-objective interval-valued programming problems, relying upon interval arithmetic rules, simplified immune response mechanisms and additive genetic operators. The main contribution of the present work
* Corresponding author.
E-mail address: [email protected] (Z. Zhang). https://doi.org/10.1016/j.engappai.2017.10.004 Received 16 December 2016; Received in revised form 29 August 2017; Accepted 9 October 2017 Available online 5 November 2017 0952-1976/© 2017 Elsevier Ltd. All rights reserved.
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includes three points: (i) two interval-based computational models are designed to identify the importance and diversity of individual for a given population, which can help us solve the difficulty of individual evaluation in interval-valued environments, (ii) a competitive and hybrid multi-objective optimization algorithm is proposed to solve MIVP, which borrows some biological inspirations from immunology and genetics, and (iii) the current work is helpful for opening the door to investigate hybrid multi-objective interval-valued optimization approaches. It is worth pointing out that, although MIIGA, like other immune genetic algorithms, is designed based on several immune metaphors from the clonal selection principle and a few inspirations presented in the genetics and evolution theory, it differs from any existing multiobjective optimization algorithms because of different problems solving and biological inspirations. Especially, whereas multi-objective immune genetic algorithms have been well studied for static multi-objective optimization problems (Luo et al., 0000), they cannot be directly applied to the case of MIVP, for which the main reason is that, on one hand, each individual should be evaluated based on interval comparison instead of real number, and on the other hand their strategies of selection, individual comparison and population diversity do not suite to multiobjective interval-valued environments. MIIGA is designed specially for MIVP problems rather than static multi-objective problems. It needs not only a computational model suitable for interval-valued environments to decide the importance of each individual in a given population, but also a new crowding degree model to measure the similarity of individual. It should be emphasized that the operators of crossover and mutation from the inspirations of genetics and evolution are only used additively to promote the process of solution search. Additionally, we also notice that immune inspirations are rarely borrowed to design optimizers for MIVP problems. The remaining text in this paper is organized as follows: Section 2 gives a survey of related work concerning evolutionary algorithms and immune optimization for MIP. In Section 3, we describe the multiobjective interval-valued programming problem to be solved. A crowding degree model is developed in Section 4. MIIGA is formulated and designed in detail in Section 5. Further, the computational complexity of MIIGA is analyzed, while three performance criteria are cited to execute algorithm comparison in Section 6. Sections 7 and 8 present the whole experimental study, including the experimental setup, test problems, experimental analysis and an engineering application. We conclude that MIIGA is a competitive optimization tool for MIVP in Section 9.
process industry systems, and gave a sufficient and necessary condition which a feasible solution existed, but did not design a solution search procedure to solve one such multi-objective problem. About model transformation on MIVP, midpoint and radius functions are usually used to change MIVP as a deterministic multi-objective programming problem. For instance, Yokota et al. (1995) made an early contribution to transforming a multi-objective interval-valued integer programming problem into a static single-objective integer programming one. The existing intelligent approaches solving MIVP models can be divided into two broader classes, i.e., model transformation approaches (Li et al., 2010; Cheng et al., 2014; Cheng et al., 2004; Yokota et al., 1995; Li et al., 2011; Bhunia and Samanta, 2014; Jiang et al., 2008; Jiang, 2005; Chen and Chen, 2014) and direct methods (Limbourg and Aponte, 2005; Zhang et al., 2008a; Sun et al., 2013; Zhang et al., 2008b; Zhang et al., 2014; Wang 2014). The former solves MIVP problems by integrating some representative multi-objective evolutionary algorithms (e.g., NSGA-II) with classical programming methods or RBF neural networks. The latter is an interval-valued intelligent optimization approach solving MIVP problems directly, for which the main challenge includes two aspects: (i) discrimination between individuals, namely identification whether an individual is superior to another one, and (ii) crowding degree models. Limbourg and Aponte (2005) based on the basic algorithmic structure of NSGA-II, suggested an extended multiobjective evolutionary algorithm (IP-MOEA) capable of being applied to MIVP problems directly. They designed an individual dominance model by using the left and right endpoints of sub-objective intervalvalued functions, while a hybrid model was developed to measure the similarity of an individual to other individuals in a given population. Additionally, studies on multi-objective evolutionary algorithms, based on possibility degree models have achieved initial success. Gong et al. (2010) and Zhang et al. (2008b) proposed two valuable multi-objective optimization algorithms related to either NSGA-II or particle swarm optimization (PSO), in which some interesting models of probabilistic dominance and crowding degree were designed to identify those valuable individuals and compute their similarity degrees. In order to find the most preferred solutions for MIVP problems, Sun et al. (2013) designed a metric index to emphasize the approximation relationship between candidate solutions and Pareto optimal solutions, and then developed an interactive evolutionary algorithm after incorporating an optimization-cum-decision-making procedure and the idea of preference selection. Zhang et al. (2008a) studied a multi-objective particle swarm optimization algorithm to handle MIVP problems without interval constraints. In one such algorithm, a probable dominance method, used in processing intervals was proposed to identify the qualities of solutions.
2. Related research work The reported studies on MIP concentrate mainly on model transformation and intelligent methodologies. Usually, several conventional approaches, e.g., interval analysis, midpoint and radius methods, weighted coefficient approaches, interval possibility models techniques and so on (Bhunia and Samanta, 2014; Cheng et al., 2014; Cheng et al., 2004; Li et al., 2011; Jiang et al., 2008; Jiang, 2005; Li et al., 2010; Yokota et al., 1995), are adopted to change uncertain objective and constraint functions into deterministic ones. Correspondingly, some achievements on model transformation have appeared in the literature. For example, Li et al. (2010) studied early how to transform MINP into a nested MIVP model by the upper and lower bounds of the objective function and constraints, and subsequently one such model was changed into a static single-objective programming model by means of the penalty function method, possibility models and midpoint and radius functions of its sub-objective interval-valued functions. This is a usual and popular model transformation method. Based on this idea of transformation, Cheng et al. (2014) discussed how MINP models were converted into unconstrained multi-objective programming ones, where the values of the uncertain objective and constraint functions for each candidate were decided by RBF neural networks, instead of Taylor series and interval analysis. Cheng et al. (2004) developed a general MINP model for
3. Problem formulation and basic concepts Let 𝐼𝑅 represent the set of bounded and closed intervals in 𝑅. Each element in 𝐼𝑅 is said to be an interval number. Consider the following general nonlinear multi-objective interval-valued programming problem of form MIVP: min 𝑓 (𝐱, 𝑈 ) = (𝑓1 (𝐱, 𝑈 ), 𝑓2 (𝐱, 𝑈 ), … , 𝑓𝑚 (𝐱, 𝑈 )), 𝐱∈𝐷
with bounded and closed domain 𝐷 in 𝑅𝑝 , and decision vector 𝐱 ∈ 𝐷, where 𝑈 denotes a 𝑞-dimensional interval vector, i.e., the product of 𝑞 interval numbers; 𝑓 (𝐱, 𝑈 ) is the interval-valued vector function. 𝑓𝑖 (𝐱, 𝑈 ) is the ith sub-objective function with 1 ≤ 𝑖 ≤ 𝑚, represented by [𝑓𝑖 (𝐱), 𝑓 𝑖 (𝐱)] by means of interval arithmetic rules. Interval analysis is a valuable tool in the fields of interval programming and dynamic systems. Moore et al. (2009) introduced some basic properties on interval number. They gave the following partial order relation between two interval numbers 𝐴 = [𝐴𝐿 , 𝐴𝑅 ] and 𝐵 = [𝐵 𝐿 , 𝐵 𝑅 ], 𝐴 < 𝐵 ⟺ 𝐴𝑅 < 𝐵 𝐿 . 236
(1)
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Fig. 1. The relationship between hypercubes of two candidates.
We here introduce the following weak comparison relation between 𝐴 and 𝐵, 𝐴 ≤ 𝐵 ⟺ 𝐴𝑅 ≤ 𝐵 𝐿 , 𝐴 ≥ 𝐵 ⟺ 𝐵 ≤ 𝐴.
include non-intersection, intersection and inclusion. In the above figure, 𝑋 and 𝑌 denote the hypercubes of interval objective vectors 𝑓 (𝐱, 𝑈 ) and 𝑓 (𝐲, 𝑈 ) for candidates 𝐱 and 𝐲, respectively; A and B represent the vertexes of 𝑋 and 𝑌 at the top-left corner, respectively; 𝑉1 and 𝑉2 stand for the hypervolumes of the corresponding dash hypercubes. Let 𝑉𝑥𝑦 be the hypervolume of the outer hypercube. In this way, 𝑉1 , 𝑉2 and 𝑉𝑥𝑦 can be represented by the endpoints of the interval-valued sub-objectives,
(2)
Generally speaking, it is not true that 𝐴 − 𝐴 = 0, for example when A=[−2,2], we can see that A-A=[−4,4]. Thus, we here prescribe that 𝐴 equals 𝐵 if and only if 𝐴𝐿 = 𝐵 𝐿 and 𝐴𝑅 = 𝐵 𝑅 . This way, Eq. (2) indicates that once 𝐴 ≤ 𝐵 and 𝐵 ≤ 𝐴, 𝐴 and 𝐵 will degenerate to 𝑎, i.e., [𝑎, 𝑎]. It is emphasized that, when 𝐴 ∩ 𝐵 ≠ 𝜙 and 𝐴 ≠ 𝐵, Eq. (2) cannot be adopted to execute comparison between 𝐴 and 𝐵. In such case, the version of possibility degree is an alternative way to compare 𝐴 with 𝐵. We here cite the following possibility degree model (Zhou and Liu, 2008), 𝑑(𝐴, 𝐶) 𝑃 𝑟{𝐴 ≤ 𝐵} = , 𝑑(𝐴, 𝐶) + 𝑑(𝐵, 𝐶)
𝑉1 = 𝑉2 =
|𝑓𝑖 (𝐱) − 𝑓𝑖 (𝐲)|, |𝑓𝑖 (𝐱) − 𝑓𝑖 (𝐲)|,
𝑉𝑥𝑦 =
(7)
(𝑐𝑖 − 𝑑𝑖 ),
𝑖=1
(3) where 𝑐𝑖 = max{𝑓𝑖 (𝐱), 𝑓𝑖 (𝐲)} and 𝑑𝑖 = min{𝑓𝑖 (𝐱), 𝑓𝑖 (𝐲)}. Additionally, the distance between A and B is defined by √ √𝑚 √∑ (8) 𝑑(𝐱, 𝐲) = √ (𝑚(𝑓𝑖 (𝐱, 𝑈 )) − 𝑚(𝑓𝑖 (𝐲, 𝑈 )))2 , 𝑖=1
where 𝑚(𝑓𝑖 (𝐱, 𝑈 )) denotes the midpoint of interval 𝑓𝑖 (𝐱, 𝑈 ). Although Eq. (8) can present the degree of approximation of points A and B, it cannot completely reflect the relation between the two hypercubes. Once we move the positions of 𝑋 and 𝑌 , the values of 𝑉1 , 𝑉2 , 𝑉𝑥𝑦 and 𝑉1 +𝑉2 𝑑(𝐱, 𝐲) will make change correspondingly, but 2𝑉 only does minor 𝑥𝑦 change. This way, the crowding degree of 𝐱 and 𝐲 in the objective space can be defined here by
(5)
𝐱∗
Naturally, is said to be a Pareto optimal solution of the above MIVP if there does not exist 𝐲 in 𝐷 such that 𝐲 ≺ 𝐱∗ . In order to identify whether one individual is superior to another one in a given population 𝐴 with size 𝑁, we introduce the following computational model to compute the rank of individual 𝐱 in 𝐴, 𝑅(𝐱) = 𝑁 − |{𝐲 ∈ 𝐴|𝐲 ≺ 𝐱}|,
𝑖=1 𝑚 ∏
𝑖=1 𝑚 ∏
where 𝐶 = [𝐶 𝐿 , 𝐶 𝑅 ], 𝐶 𝑅 = max Γ and 𝐶 𝐿 = max Γ ⧵ {𝐶 𝑅 } with Γ = {𝐴𝐿 , 𝐴𝑅 , 𝐵 𝐿 , 𝐵 𝑅 }, and meanwhile √ (𝐴𝐿 − 𝐶 𝐿 )2 + (𝐴𝑅 − 𝐶 𝑅 )2 𝑑(𝐴, 𝐶) = . (4) 2 Based on Eq. (3), Zhou and Liu (2008) introduced the version of individual dominance. In other words, 𝐱 dominates 𝐲 (𝐱 ≺ 𝐲) if 𝜎𝑖 (𝐱, 𝐲) ≥ 𝜎𝑖 (𝐲, 𝐱) for any 𝑖 with 1 ≤ 𝑖 ≤ 𝑚, and there exists 𝑖0 such that 𝜎𝑖0 (𝐱, 𝐲) > 𝜎𝑖0 (𝐲, 𝐱), where 𝜎𝑖 (𝐱, 𝐲) = 𝑃 𝑟{𝑓𝑖 (𝐱, 𝑈 ) ≤ 𝑓𝑖 (𝐲, 𝑈 )}.
𝑚 ∏
𝐷(𝐱, 𝐲) =
𝑉1 + 𝑉2 𝑑(𝐱, 𝐲). 2𝑉𝑥𝑦
(9)
𝑉 +𝑉
Again, since 0 ≤ 1𝑉 2 < 1, 𝐷(𝐱, 𝐲) changes within 0 and 𝑑(𝐱, 𝐲). Eq. 𝑥𝑦 (9) hints that if 𝑋 and 𝑌 are close, 𝐷(𝐱, 𝐲) is small. Therefore, in order to reflect the diversity of a given population 𝑃 , the crowding degree of individual 𝐱 in 𝑃 is designed by
(6)
where |Λ| represents the number of elements in set Λ. Eq. (6) indicates that 𝐱 is better if 𝑅(𝐱) is larger, and especially it is a non-dominated individual in 𝐴 if 𝑅(𝐱) = 𝑁. This way, 𝐴 may be divided into two subpopulations. One is composed of non-dominated individuals in 𝐴, the other consists of those dominated ones.
𝐷(𝐲, 𝐱) + 𝐷(𝐱, 𝐳) (10) 2 where 𝐲 and 𝐳 are two closest individuals of 𝐱 in 𝑃 . Eq. (10) illustrates that, if the crowding degrees of individuals in 𝑃 are small, these individuals will have high similarity and thus weak population diversity. 𝐶(𝐱) =
Remark 1. When computing the ranks of all individuals in 𝐴, we first need to calculate the possibility degrees of sub-objective interval numbers for any two individuals through Eq. (3), for which the total of executions is 𝑚𝑁 2 . Second, we also execute 𝑚𝑁 2 comparisons between individuals through non-dominated sorting. Once all the non-dominated individuals in 𝐴 need to be discriminated, the computational complexity is 𝑂(𝑚𝑁 2 ).
Remark 2. In order to compute the crowding degrees of individuals in 𝑃 with size 𝑁, the two closest individuals of each individual in the objective space are first decided, for which we need to execute ( 12 𝑁(𝑁 − 1) + 𝑁𝑙𝑜𝑔𝑁) times. Thus, once Eq. (10) is enforced in 𝑃 , it will cause the complexity of 𝑂(𝑁 2 ). 5. Algorithm formulation and design
4. Crowding degree model Based on the above designs, immune inspirations and also additive genetic operators, we in this section formulates MIIGA, for which the flowchart is given in Fig. 2. One such approach includes three functional modules, i.e., population division, immune evolution and
As we know, the objective interval vectors of individuals 𝐱 and 𝐲, i.e., 𝑓 (𝐱, 𝑈 ) and 𝑓 (𝐲, 𝑈 ), correspond to two hypercubes in the objective space. In general, their possible position relations, given by Fig. 1 237
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high-quality and diverse antibodies. More precisely, after computing the ranks of antibodies in 𝐴 through Eq. (6), population 𝐵 here consists of elements in 𝐴 with the largest rank, namely 𝐵 = {𝐱 ∈ 𝐴|𝑅(𝐱) = max{𝑅(𝐳)|𝐳 ∈ 𝐴}},
(11)
and meanwhile 𝐶 is composed of the elements in 𝐴 but not in 𝐵. (ii) Proliferation & Polynomial(𝐵). Each antibody in 𝐵 proliferates 𝑚𝑐 clones, while changing its genes with mutation rate 𝑝𝑚 through polynomial mutation; subsequently, all the mutated clones constitute clonal population 𝐵 ∗ . (iii) Proliferation & Nonuniform(𝐶). Each antibody in 𝐶 reproduces (𝑚𝑐 − 1) clones. Each clone mutates its genes with mutation rate 𝑝𝑚 through nonuniform mutation; all the mutated clones forms clonal population 𝐶 ∗ . (iv) Selection(𝐻). Such module first computes the ranks of antibodies in 𝐻 through Eq. (6). Second, those non-dominated antibodies constitute a temporary population 𝐷. If |𝐷| > 𝑁, Eq. (10) is used to delete a part of antibodies in 𝐷 with small crowding degrees, and otherwise, some dominated antibodies with large crowding degrees are selected to insert into 𝐷. (v) Recruitment(𝐷). This denotes simply that 𝜇% of elements in 𝐷 are replaced by new antibodies generated randomly. In other words, if 𝐷 only consists of non-dominated antibodies, those elements with small crowding degrees are substituted by some new random antibodies, and otherwise, such new antibodies displace those dominated elements with high similarity. ∗ ). This only keeps those non-dominated (vi) Elitism (𝑀𝑠𝑒𝑡 ∪ 𝑀𝑠𝑒𝑡 memory cells in terms of Eq. (6). Once the number of such cells is beyond 𝑀𝑚𝑎𝑥 , those redundant elements with small crowding degrees vanish from the memory pool by means of Eq. (10).
Fig. 2. The flowchart of MIIGA.
genetic evolution. The current population is divided into two subpopulations; one consists of high-quality individuals, the other is composed of other individuals. The first sub-population not only updates the memory pool, but also participates in evolution through proliferation and polynomial mutation; the second sub-population seeks different kinds of new individuals through nonuniform mutation. The memory pool is mainly used to accelerate its remembers to discover high-quality individuals. Additionally, some random individuals are produced to adjust the diversity of population. In this algorithm statement, we view antigen 𝐴𝑔 as MIVP itself; candidate solutions from the design space 𝐷 are regarded as real-coded antibodies; memory cells are viewed as those best antibodies found until now. Following these descriptions and Fig. 2, MIIGA can be described as given in the form of table at the top of next page. In the above formulation, steps 5.1 to 5.5 are a period of evolution for searching a part of Pareto optimal solutions in MIVP; steps 5.1 and 5.2 are required to determine two sub-populations by means of the ranks of antibodies in 𝐴; step 5.3 produces a new evolving population through immune evolution and population update. Apart from eliminating redundant memory cells, step 5.4 urges those memory cells to discover high-quality antibodies through genetic evolution. More precisely, step 5.4.1 directly eliminates identical or dominated elements, after combining the existing memory cells with those remembers in 𝐵; step 5.4.2 makes the memory set 𝑀𝑠𝑒𝑡 create new antibodies through arithmetic crossover and nonuniform mutation with crossover rate 𝑝𝑐 and mutation rate 𝑝𝑚 ; step 5.4.3 collects those elitist memory cells found. Within a period of iteration, besides the current population is responsible for exploring excellent and diverse antibodies through evolution, a few new random antibodies are inserted to maintain the diversity of population. This cannot only avoid the population to get into local search, but also is helpful for strengthening the ability of global evolution. In order to accelerate the progress of finding diverse Pareto optimal solutions, MIIGA utilizes the additive genetic operators to make those memory cells move towards the desired regions as possible, which can enhance the quality of solution search.
6. Computational complexity and performance criteria 6.1. Computational complexity In this section, MIIGA’s computational complexity is analyzed in order to know which factors influence its performance efficiency. Theorem 1. In the worst case, the computational complexity of MIIGA is 𝑂(𝑁𝑝𝑚𝑐 + (𝑁 + 𝑀𝑚𝑎𝑥 )2 𝑙𝑜𝑔(𝑁 + 𝑀𝑚𝑎𝑥 )).
5.1. Illustration on main modules
Proof. Through MIIGA’s formulation, step 5.1 computes the ranks of antibodies in 𝐴 with the complexity of 𝑂(𝑁 2 ) mentioned in Remark 1. The computational complexity of step 5.3 are decided by Steps 5.3.1 and 5.3.4. To this point, step 5.3.1 produces 𝑁𝑚𝑐 clones by proliferation, among which each clone mutates its genes with at most 𝑝 times. Consequently, the complexity of one such step is 𝑂(𝑁𝑝𝑚𝑐 ) in the worst case. Like step 5.1, step 5.3.4 needs to decide the ranks of antibodies in 𝐻 with the complexity of 𝑂(|𝐻|2 ); once all elements in 𝐻 are nondominated, we will have to compute their crowding degrees with the complexity of 𝑂(|𝐻|2 ) by Remark 2. Since |𝐻| ≤ 2𝑁, the complexity of step 5.3 in the worst case is 𝑂(𝑁𝑝𝑚𝑐 + 𝑁 2 ). On the other hand, the computational complexity of step 5.4 is determined by steps 5.4.1 and 5.4.3. After the memory pool in step 5.4.1 collects those elitist antibodies in 𝐵, those dominated members in 𝑀𝑠𝑒𝑡 will be deleted through Eq. (6), and thus the complexity is at most 𝑂((𝑁 + 𝑀𝑚𝑎𝑥 )2 ) because of |𝐵| ≤ 𝑁. Further, Step 5.4.3 first eliminates those dominated memory cells in ∗ with the same complexity as that in step 5.4.1 because 𝑀𝑠𝑒𝑡 ∪ 𝑀𝑠𝑒𝑡 ∗ |. Afterwards, when the number of non-dominated cells |𝑀𝑠𝑒𝑡 | = |𝑀𝑠𝑒𝑡 is beyond 𝑀, Eq. (10) is used to delete those memory cells with small ∗ |2 𝑙𝑜𝑔|𝑀 ∪ crowding degrees, for which the complexity is 𝑂(|𝑀𝑠𝑒𝑡 ∪𝑀𝑠𝑒𝑡 𝑠𝑒𝑡 ∗ |). Since |𝑀 | ≤ 𝑁 + 𝑀 𝑀𝑠𝑒𝑡 𝑠𝑒𝑡 𝑚𝑎𝑥 in such case, the complexity is at most 𝑂((𝑁 + 𝑀𝑚𝑎𝑥 )2 𝑙𝑜𝑔(𝑁 + 𝑀𝑚𝑎𝑥 )). Summarily, MIIGA’s computational complexity in the worst case is decided by
(i) Division(𝐴). This divides population 𝐴 into two sub-populations of 𝐵 and 𝐶 so that different evolution fashions can be adopted to explore
𝑂 = 𝑂(𝑁 2 ) + 𝑂(𝑁𝑝𝑚𝑐 + 𝑁 2 ) + 𝑂((𝑁 + 𝑀𝑚𝑎𝑥 )2 𝑙𝑜𝑔(𝑁 + 𝑀𝑚𝑎𝑥 )) = 𝑂(𝑁𝑝𝑚𝑐 + (𝑁 + 𝑀𝑚𝑎𝑥 )2 𝑙𝑜𝑔(𝑁 + 𝑀𝑚𝑎𝑥 )). 238
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The proposed approach MIIGA 1. Input: Population size 𝑁, maximal memory size 𝑀𝑚𝑎𝑥 , clonal size 𝑚𝑐 , crossover rate 𝑝𝑐 , mutation rate 𝑝𝑚 and maximal iterative number 𝐺𝑚𝑎𝑥 ; 2. 0 ← 𝑛, 𝑀𝑠𝑒𝑡 ← 𝜙; 3. Generate an initial antibody population 𝐴 of 𝑁 random antibodies; 4. Evaluate antibodies in 𝐴 by interval arithmetic rules; 5. While 𝑛 < 𝐆𝑚𝑎𝑥 do 5.1. Compute the ranks of the antibodies in 𝐴 by Eq. (6); 5.2. Execute population division, (𝐵, 𝐶): =Division(𝐴); % Superior and inferior sub-populations 5.3. Carry out immune evolution on 𝐴: % New high-quality and diverse antibodies 5.3.1. 𝐵 ∗ ∶=Proliferation & Polynomial(𝐵); % Proliferation and Polynomial mutation 5.3.2. 𝐶 ∗ ∶= Proliferation & Nonuniform(𝐶); % Proliferation and nonuniform mutation 5.3.3. 𝐻 ∶= 𝐵 ∪ 𝐵 ∗ ∪ 𝐶 ∗ ; % Population combination 5.3.4. Implement immune selection, 𝐷=Selection(𝐻); % Elitist and diverse antibodies 5.3.5. Create new population, 𝐴=Recruitment(𝐷); % Diversity maintenance 5.4. Enforce genetic evolution on 𝑀𝑠𝑒𝑡 : % Acceleration on solution search 5.4.1. Collect elitism, 𝑀𝑠𝑒𝑡 =Update(𝑀𝑠𝑒𝑡 , 𝐵); % Elitist memory cells ∗ 5.4.2. Perform genetic evolution, 𝑀𝑠𝑒𝑡 =Evolution(𝑀𝑠𝑒𝑡 ); % Crossover and mutation ∗ =Elitism (𝑀 ∗ 5.4.3. Transact memory update,𝑀𝑠𝑒𝑡 𝑠𝑒𝑡 ∪ 𝑀𝑠𝑒𝑡 );% Elitist memory cells 5.5. 𝑛 ← 𝑛 + 1; 6. End while; 7. Output: Pareto optimal solutions in 𝑀𝑠𝑒𝑡 .
are with high quality but also with high similarity and narrow coverage scope; when the second case occurs, X is poor. Thus, when evaluating the quality of X, the following indices should be included. (ii) Coverage width 𝐶𝑊 . This denotes the Hausdorff distance of the two farthest hypercubes over the non-dominated front of solutions in 𝑋, namely (13)
𝐶𝑊 = max max 𝐻(𝑓𝑖 (𝐱, 𝑈 ), 𝑓𝑖 (𝐲, 𝑈 )). 𝐱,𝐲∈𝑋 1≤𝑖≤𝑚
(iii) Solution density (𝑆𝐷). This is used to measure the distribution characteristic of solutions in 𝑋 in the objective space, defined by √ ∑|𝑋| 1 𝑆𝐷 = (𝑠 − 𝑠𝑖 )2 , 𝑖=1 (14) |𝑋| − 1 𝑠𝑖 = min{𝑠(𝑓 (𝐱𝑖 , 𝑈 ), 𝑓 (𝐱𝑗 , 𝑈 ))|𝐱𝑖 , 𝐱𝑗 ∈ 𝑋, 𝑗 ≠ 𝑖}, where 𝑠 denotes the average of 𝑠𝑖 with 1 ≤ 𝑖 ≤ |𝑋|, and 𝑠(𝐱𝑖 , 𝐱𝑗 ) is given by
Fig. 3. Schematic illustration on how to compute the hypervolume of the hypercone enclosed by the origin and the midpoints of non-dominated hypercubes.
𝑠(𝐱𝑖 , 𝐱𝑗 ) = 1+ Through the above inference, we notice that MIIGA’s computational complexity depends on the parameters of 𝑀𝑚𝑎𝑥 , 𝑁, 𝑚𝑐 and 𝑝. This indicates that it is difficult to solve MIVP, due to high computational complexity. Especially, once 𝑁𝑝𝑚𝑐 ≪ 𝑀𝑚𝑎𝑥 , the complexity will be 2 𝑙𝑜𝑔 𝑀 decided by 𝑂(𝑀𝑚𝑎𝑥 𝑚𝑎𝑥 ). This is why we develop an immune genetic algorithm with small population.
𝑔(𝑖, 𝑗, 𝑙) = 2
∑𝑚 𝑙=1
𝑒−𝑔(𝑖,𝑗,𝑙)
1 , √ (𝑓 (𝐱𝑖 ) − 𝑓 (𝐱𝑗 ))2 + (𝑓 𝑙 (𝐱𝑖 ) − 𝑓 𝑙 (𝐱𝑗 ))2 𝑙
𝑙
𝑚 ∑ max{0, min{𝑤𝑙 (𝑖, 𝑗), 𝑤𝑙 (𝑗, 𝑖), 𝑤𝑙 (𝑖, 𝑖), 𝑤𝑙 (𝑗, 𝑗)}} , 𝑤𝑙 (𝑖, 𝑖) + 𝑤𝑙 (𝑗, 𝑗) 𝑙=1
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𝑤𝑙 (𝑖, 𝑗) = 𝑓 𝑙 (𝑥𝑖 ) − 𝑓 (𝐱𝑗 ), 𝑤𝑙 (𝑗, 𝑖) = 𝑓 𝑙 (𝑥𝑗 ) − 𝑓 (𝐱𝑖 ). 𝑙
𝑙
Eq. (14) indicates that if 𝑆𝐷 is small, the solutions, presented in 𝑋 are with uniform distribution, and conversely such solutions are with poor distribution. (iv) Uncertain degree (𝑈 𝐷). This measures whether the hypercones of all elements in 𝑋 in the interval-valued objective space are small. It is simply defined as the total of hypervolumes of the elements in the objective space. If 𝑈 𝐷 is small, such elements are said to be with low uncertainty. Through the designs of the above indices, if algorithm 𝐴 can acquire smaller values on 𝐻𝑀, 𝑆𝐷 and 𝑈 𝐷 as well as a larger one on 𝐶𝑊 than algorithm 𝐵, the former is better.
6.2. Performance criteria Since the Pareto front of a given MIVP problem consists of a series of hypercubes, the reported performance criteria, used in comparing static multi-objective intelligent approaches is not suitable for multi-objective interval-valued intelligent approaches. In order to examine MIIGA’s performance, we here improve and cite three reported performance criteria (Gong et al., 2010; Wang, 2014) to measure the quality of a given finite solution set 𝑋. (i) Hypervolume measure (HM ). 𝐻𝑀, illustrated by Fig. 3 above represents the hypervolume of the hypercones enclosed by the thick lines and the origin. Since the Pareto front of a given MIVP problem is composed of a series of hypercubes rather than a series of points, in the case where HM is small, it is possible that two cases will take place, namely (a) all solutions in X are close to the Pareto front in the objective space, and (b) X includes fewer points, while these points are similar and far from the Pareto front. The first case shows that all the solutions in X
7. Experimental study Our experiments are executed on a computer with CPU/3.00 GHz and RBM/2.00 GB by means of Visual C++ platform. As associated to the following six multi-objective interval-valued test problems acquired through modifying the reported static benchmark problems, the proposed approach, MIIGA as in Section 5, is compared against 239
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(a) KURI .
(b) POLI .
(c) ZDT1I .
(d) ZDT2I .
(e) ZDT3I .
(f) ZDT4I .
Fig. 4. Comparison of box plot on 𝐻𝑀.
three competitive intelligent optimizers found, namely two multiobjective evolutionary algorithms (G-MOEA (Gong et al., 2010) and IP-MOEA (Limbourg and Aponte, 2005)) and one multi-objective particle swarm optimization algorithm so-called FD-MOPSO (Zhang et al., 2014). These compared approaches are specially designed to solve MIVP problems, based on the conventional multi-objective intelligent optimization approaches. Additionally, the existing static multi-objective intelligent approaches are usually only suitable for static multi-objective problems. In order to examine whether one such type of algorithm can solve MIVP problems, we take two static multi-objective approaches, i.e., multi-objective genetic algorithm NSGA-II (Deb et al., 2002) and multi-objective immune genetic algorithm MIGA (He et al., 2010) for example to handle the static multi-objective problems in which their sub-objectives are the midpoint functions of MIVP’s interval-valued sub-objectives. In order to ensure the fairness of comparison, all the approaches terminate their solution search procedures only when their respective iterative number is 500, while each approach executes 100 single runs on each test example. The settings of parameters for the compared approaches are the same as those presented in the corresponding literatures, for example, IP-MOEA and G-MOEA takes their population sizes as 50, and FDMOPSO’s population size is defined as 10. Through the above analysis on computational complexity as in Section 6, 𝑁, 𝑚𝑐 and 𝑀𝑚𝑎𝑥 , included in MIIGA are three crucial parameters to influence the efficiency of solution search. In order that MIIGA takes a trade-off between efficiency and effect, 𝑁 and 𝑚𝑐 are required to take small values as possible, i.e., 𝑁 ∈ [5, 10] and 𝑚𝑐 ∈ [2, 10]. In addition, 𝑀𝑚𝑎𝑥 is the size of the outer archive set for storing those elitist solutions acquired, it takes a somewhat large value, i.e., 𝑀𝑚𝑎𝑥 ∈ [50, 80]. After manually
experimental tuning, we take 𝑁 = 10, 𝑚 = 50, 𝑚𝑐 = 2, 𝑝𝑐 = 0.45 and 𝑝𝑚 = 0.1. 7.1. Multi-objective interval-valued test problems Limbourg and Aponte (2005) proposed a valuable approach that transformed static multi-objective benchmark problems into multiobjective interval-valued programming ones, in which a disturbed vector-valued function was designed to add to static objective functions. More precisely, let 𝛿(𝐱) be a two-dimensional vector-valued function, 𝛿(𝐱) = (cos(10𝜋
𝑛 ∑ 𝑖=1
𝑥𝑖 )∕2, cos(20𝜋
𝑛 ∑
𝑥𝑖 )∕2)𝑇 .
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𝑖=1
In this way, the following static multi-objective problem min 𝑓 (𝐱) = (𝑓1 (𝐱), 𝑓2 (𝐱), … , 𝑓𝑚 (𝐱)) 𝐱∈𝐷
can be changed as the multi-objective interval-valued problem MIVP in Section 3, where 𝑓𝑖 (𝐱, 𝛿) = [𝑓 (𝐱, 𝛿), 𝑓 𝑖 (𝐱, 𝛿)], with 𝑓 (𝐱, 𝛿) = 𝑖
𝑖
min{𝑓𝑖 (𝐱), 𝑓𝑖 (𝐱) + 𝛿1 (𝐱)} and 𝑓 𝑖 (𝐱, 𝛿) = max{𝑓𝑖 (𝐱), 𝑓𝑖 (𝐱) + 𝛿2 (𝐱)}. 𝛿1 (𝐱) and 𝛿2 (𝐱), which change over an interval [−0.5, 0.5], are the first and second elements of 𝛿(𝐱), respectively. Based on the above problem transformation, we here transform in turn six static multi-objective benchmark problems (KUR, POL, and ZDT1 to ZDT4) (Deb et al., 2002) into multi-objective programming problems (KUR𝐼 , POL𝐼 and ZDT1𝐼 to ZDT4𝐼 ). In order to examine the ability of dealing with interval-valued programming problems for the above approaches, the dimensions of the test problems ZDT1 to ZDT3 are set as 30, but the dimension of ZDT4 is considered as 10. Among these static test problems, the two multimodal optimization problems POL and KUR with dimensions 2 and 240
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(a) KURI .
(b) POLI .
(c) ZDT1I .
(d) ZDT2I .
(e) ZDT3I .
(f) ZDT4I .
Fig. 5. Comparison of box plot on 𝐶𝑊 .
Figs. 4 and 5 show that the above six approaches have different solution qualities and search stability in terms of their quartiles. We notice that NSGA-II and MIGA as two static multi-objective intelligent approaches result in poor search performances by comparison with the other four approaches designed specially for MIVP problems. Whereas NSGA-II can get smaller means on 𝐻𝑀 for all the test examples, so are its means on 𝐶𝑀, which means that one such approach can only find some solutions with high similarity and narrow coverage scope and meanwhile it gets easily into local search. MIGA can obtain some solutions with relatively wide coverage scope for some test problems by Figs. 5 and 8, and thus is with strong population diversity, whereas its solution qualities are both poor and instable by Fig. 4. The main reason involves two points: (a) it can only find some poor solutions for KURI and instable solution qualities for the other test problems because of wide regions between the upper and lower quartiles, and (b) it produces many singular points outside its upper and lower quartiles inFig. 4, which hints that its solution quality acquired for each test problem cannot keep stable. IP-MOEA and G-MOEA perform poor over FD-MOPSO and MIIGA, for which the main reason includes three points: (a) IP-MOEA results in the serious phenomenon of fluctuation of solution search for the test examples except for ZDT4I , while some singular points in its box plots are found. Thereby, by comparison against G-MOEA its solution qualities for the test problems are poor but superior to those acquired by NSGA-II and MIGA, and (b) although G-MOEA can acquire satisfactory solution qualities for POLI , ZDT2I and ZDT3I , it produces some singular points for some test problems, and meanwhile the hypercubes of its solutions for the test problems are with relatively high uncertainty which can be known below.
3 respectively include multiple noncontinuous Pareto front segments. In addition, the test problems ZDT1 to ZDT4 are a set of typical and difficult benchmark problems with different attributes. ZDT1 and ZDT2 have similar characteristics, namely their Pareto fronts are continuous, smooth, and convex for ZDT1 but concave for ZDT2. The Pareto front of ZDT3 consists of 5 disconnected curve segments. Note that the reason why ZDT4 here takes dimension 10 instead of 30, is that it includes 219 local Pareto fronts but only one global Pareto front and hence it is extremely difficult to search the global Pareto front. Since all these static problems are hard and 𝛿1 (𝐱) and 𝛿2 (𝐱) are two uncertain factors, their corresponding multi-objective interval-valued programming problems can be selected to examine the performance characteristics of the above algorithms. 7.2. Experimental results and analysis After respectively executing 100 runs on each interval-valued problem above, the algorithms acquire their respective solution sets. Correspondingly, their statistical characteristics, based on the performance criteria in Section 7 are presented by box-plots through Figs. 4–7. Additionally, we only take KUR𝐼 for example to present the solution distribution in the rectangle objective space, due to the limit of page space. The non-dominated fronts formed by the rectangles of the objective interval vectors, gotten by them after respectively a single run are drawn in Fig. 8. The experimental results, presented by box plots in Figs. 4–8 expose some significant differences between the above four approaches. We can draw the following conclusions: 241
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(a) KURI .
(b) POLI .
(c) ZDT1I .
(d) ZDT2I .
(e) ZDT3I .
(f) ZDT4I .
Fig. 6. Comparison of box plot on 𝑆𝐷. Table 1 Comparison on average runtime (second) spent by each approach. Below A, B, C, D, E and F denote orderly by IP-MOEA, G-MOEA,FD-MOPSO,NSGA-II,MIGA and MIIGA.
By comparing MIIGA with FD-MOPSO, the former is superior to the latter for KURI , POLI and ZDT4I in terms of Figs. 4 and 5. Additionally, whereas MIIGA gains some relatively larger means on 𝐻𝑀 than FDMOPSO for ZDT1I , ZDT2I and ZDT3I by Fig. 4, by Fig. 5 it acquires wider coverage scopes and low uncertainties of solutions (see Fig. 7) for such problems. Therefore, MIIGA is globally superior to FD-MOPSO with the aspect of solution quality (also see Fig. 8). Fig. 6 illustrates that the non-dominated solutions, acquired by MIIGA per run for east test problem are with satisfactory and stable solution distribution; IP-MOEA, G-MOEA and FD-MOPSO are secondary. NSGA-II and MIGA can only find some solutions with poor distribution and distributional instability. Fig. 7 hints that the non-dominated solutions, gotten by NSGA-II and MIGA are with high uncertainty, namely the widths of hypercubes of the solutions are large, and thereby such two approaches as static optimization ones cannot gain satisfactory effects for MIVP problems. Additionally, the non-dominated solutions, found by IP-MOEA and G-MOEA for the test problems have relatively large hypervolumes, and consequently cause relatively high uncertainties but superior to those gotten by NSGA-II and MIGA. Further, the solutions, obtained by MIIGA for ZDT1𝐼 and ZDT3𝐼 are with lower uncertainty than those acquired by FD-MOPSO, while FD-MOPSO can gain some solutions with somewhat lower uncertainty for KUR𝐼 , POL𝐼 , ZDT2𝐼 and ZDT4𝐼 . Thereby, FD-MOPSO is somewhat superior to MIIGA with regard to solution uncertainty, but it produce some large hypercubes for some runs and test examples, e.g. KUR𝐼 (see Fig. 8). With respect to performance efficiency, Table 1 illustrates that for each test problem, FD-MOPSO spends the least runtime, while Figs. 4–8
A B C D E F
KUR𝐼
POL𝐼
ZDT1𝐼
ZDT2𝐼
ZDT3𝐼
ZDT4𝐼
0.48 6.12 0.14 7.28 11.7 10.68
0.42 6.18 0.14 7.8 16 13.46
0.5 5.92 0.18 6.62 7.26 3.1
0.5 5.9 0.12 6.34 7.42 2.22
0.5 5.96 0.2 5.98 7.32 3.38
0.48 6.06 0.12 6.88 9.56 2.46
show that it performs well over IP-MOEA, G-MOEA, NSGA-II and MIGA but poor over MIIGA. Thereby, FD-MOPSO is a competitive optimizer. Whereas IP-MOEA only needs less runtime to solve each test problem by comparison with G-MOEA, NSGA-II, MIGA and MIIGA, its solution search effect is poor. MIIGA can spends less runtime to handle ZDT1I , ZDT2I , ZDT3I and ZDT4I by comparison against G-MOEA, NSGA-II and MIGA. When solving KURI and POLI , it is only more efficient than MIGA, for which the main reason consists in that such two examples are lowdimensional and easy to solve while its memory pool needs to eliminate many old memory cells within each period of evolution. Summarily, when solving the above higher-dimensional hard interval-valued problems, the six approaches present significant differences. MIIGA performs globally well over the compared approaches according to the performance criteria in Section 6.2, for which the main reason consists in that it inherits the merits of immune optimization and genetic algorithms, and meanwhile the proposed crowding degree model can produce the strong diversity of population and strengthen the ability of individual selection in the process of solution search. In 242
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(a) KURI .
(b) POLI .
(c) ZDT1I .
(d) ZDT2I .
(e) ZDT3I .
(f) ZDT4I .
Fig. 7. Comparison of box plot on 𝑈 𝐼.
Fig. 8. Comparison of non-dominated fronts acquired for the six approaches with a run for KURI .
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(a) 𝐻𝑀.
(b) 𝐶𝑊 .
(c) 𝑆𝐷.
Fig. 9. Comparison of box plot acquired by the six approaches for EED.
̄ 𝑃̄𝐿 ; Λ(𝑃𝐺 , 𝛿) = where Λ(𝑃𝐺 , 𝛿) = [Λ(𝑃𝐺 , 𝛿), Λ(𝑃𝐺 , 𝛿)] with Λ = 𝐹̄ , 𝐸, 𝑚𝑖𝑛{Λ(𝑃𝐺 ), Λ(𝑃𝐺 ) + 𝛿1 (𝑃𝐺 )}, and Λ(𝑃𝐺 , 𝛿) = 𝑚𝑎𝑥{Λ(𝑃𝐺 ), Λ(𝑃𝐺 ) + 𝛿2 (𝑃𝐺 )}. 𝛿1 (𝑃𝐺 ) and 𝛿2 (𝑃𝐺 ) are mentioned in Section 7.1. This model is solved based on the standard IEEE 30-bus 6-generator test system. All the parameter settings but 𝑁 = 6, 𝑃𝐷 = 2.834, 𝑃𝐺𝑚𝑖𝑛 = 0.05, 𝑃𝐺𝑚𝑎𝑥 = 1.5 with 𝑖 𝑖 1 ≤ 𝑖 ≤ 6 can be found in Wu (2007). The above fix approaches are then tested on the above interval programming model. Their respective maximal iterative number is 500 in a run, and meanwhile each algorithm executes 100 single runs. Like the above experiments, their box plots on indices 𝐻𝑀, 𝐶𝑊 and 𝑆𝐷 are given in Fig. 9. The box plots in Figs. 9(a) and (b) hints that MIIGA can gain the best solution quality among the six approaches while being capable of performing stable solution search; FD-MOPSO is secondary. Although G-MOEA produces several singular points, it behaves stable search performance. Nevertheless, NSGA-II and MIGA can only present the worst search performances, and thus are further confirmed to be not suitable for MIVP problems. Hence, only when being made major improvements, they are desired for MIVP. Additionally, IP-MOEA is also difficult for one such problem, as it presents the phenomenon of instable solution search. Fig. 9(c) illustrates that G-MOEA, FD-MOPSO, MIGA and MIIGA can find some solutions with stable and satisfactory distribution, whereas IP-MOEA and NSGA-II are opposite.
addition, FD-MOPSO is a competitive optimizer for MIVP. NSGA-II and MIIGA as two static optimization approaches cannot effectively solve MIVP problems even if they are competitive for static multi-objective optimization problems. 8. Engineering application Generally, the environmental/economic power dispatch problem needs to optimize simultaneously two competitive objective functions, i.e., fuel cost 𝐹 (𝑃𝐺 ) and emission 𝐹 (𝑃𝐺 ), while satisfying several constraint limits. Such two functions are defined usually by 𝐹 (𝑃𝐺 ) = 𝐸(𝑃𝐺 ) =
𝑁 ∑ (𝑎𝑖 + 𝑏𝑖 𝑃𝐺𝑖 + 𝑐𝑖 𝑃𝐺2 ), 𝑖=1 𝑁 ∑
𝑖
(17) −2
{10
𝑖=1
(𝛼𝑖 + 𝛽𝑖 𝑃𝐺𝑖 + 𝛾𝑖 𝑃𝐺2 ) + 𝜉𝑖 𝑖
exp(𝜆𝑖 𝑃𝐺𝑖 )},
where 𝑃𝐺 = (𝑃𝐺1 , 𝑃𝐺2 , … , 𝑃𝐺𝑁 ), 𝑃𝐺𝑚𝑖𝑛 ≤ 𝑃𝐺𝑖 ≤ 𝑃𝐺𝑚𝑎𝑥 , 1 ≤ 𝑖 ≤ 𝑁; 𝑁 𝑖 𝑖 is the number of generators; 𝑎𝑖 , 𝑏𝑖 and 𝑐𝑖 are the cost coefficients for the ith generator, and meanwhile 𝛼𝑖 , 𝛽𝑖 , 𝛾𝑖 , 𝜉𝑖 and 𝜆𝑖 are the emission coefficients; 𝑃𝐺𝑖 is the real power output of the ith generator. However, another reported and important function 𝑃𝐿 , line loss, is not taken as an objective function in general, given by 𝑃𝐿 (𝑃𝐺 ) =
𝑁 𝑁 ∑ ∑ 𝑗=1 𝑖=1
𝑃𝐺𝑖 𝐵𝑖𝑗 𝑃𝐺𝑗 +
𝑀 ∑
𝐵0𝑖 𝑃𝐺𝑖 + 𝐵00 ,
(18)
9. Conclusions and further work
𝑖=1
where 𝐵𝑖𝑗 , 𝐵0𝑖 and 𝐵00 are the transmission network power loss coefficients for the ith generator. This way, Wu (2007) proposed a threeobjective Environmental/Economic power dispatch (EED) programming model given by
Based on the fact that multi-objective interval-valued programming is a hard topic with comprehensive application background, this work aims at studying an immune genetic optimization approach. We first develop a model to identify the importance of individual in a given population, relying upon the reported concepts of possibility degree and interval dominance. Second, a crowding degree model is designed to determine the degree of similarity for an individual to other individuals, used in picking up diverse individuals in the population. Finally, an immune genetic evolutionary mechanism is designed to seek those high-quality solutions, in which an immune evolutionary mechanism is developed to make the current population explore those diverse and elitist individuals through proliferation, immune selection, and different kinds of mutation operations. The genetic evolutionary mechanism as an additional operator accelerates those remaining memory cells to explore the desired solutions in the design space. The proposed method, with the merits of small population and strong evolution, only includes three adjustable parameters 𝑚𝑐 , 𝑝𝑐 and 𝑝𝑚 . The theoretical analysis has showed that MIIGA’s computational cost is decided mainly by 𝑀𝑚𝑎𝑥 . Comparative experiments have demonstrated that the approach is useful in
min 𝐻(𝑃𝐺 ) = (𝐹 (𝑃𝐺 ), 𝐸(𝑃𝐺 ), 𝑃𝐿 (𝑃𝐺 )), 𝑥∈𝐷
s. t.
𝑁 ∑
𝑃𝐺𝑖 − 𝑃𝐷 − 𝑃𝐿 (𝑃𝐺 ) = 0
𝑖=1
where the equality means that the total power generation must include the real total power demand 𝑃𝐷 and power loss 𝑃𝐿 in transmission lines. When handling the equality constraint, each sub-objective function is ∑ attached here by a penalty function, i.e., | 𝑁 𝑖=1 𝑃𝐺𝑖 − 𝑃𝐷 − 𝑃𝐿 (𝑃𝐺 )|. Like the model transformation fashion in Section 7.1, by means of Eq. (16) we transform the above static three-objective optimization problem into the following three-objective interval-valued problem, ̄ 𝐺 , 𝛿), 𝑃̄𝐿 (𝑃𝐺 , 𝛿)), min 𝐻(𝑃𝐺 , 𝛿) = (𝐹̄ (𝑃𝐺 , 𝛿), 𝐸(𝑃 𝑥∈𝐷
s. t.
𝑁 ∑
𝑃𝐺𝑖 − 𝑃𝐷 − 𝑃𝐿 (𝑃𝐺 ) = 0,
𝑖=1
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practice and performs well over the compared approaches while NSGAII and MIGA as static multi-objective approaches cannot effectively deal with MIVP. In this work, we, however, only make some preliminary investigations on how to explore immune genetic optimization to solve multi-objective interval-valued programming problems. Whereas MIIGA exhibits some excellent properties, some improvements need to be made. For example, its structures will be optimized further in the precondition of improving the solution quality, and also its engineering applications will be further investigated. Additionally, we claim that MIIGA is a hybrid optimization approach which skillfully merges immune and genetic operators into its evolutionary mechanism. This is the first hybrid intelligent optimization approach for MIVP problems. Meanwhile, it is available to investigate hybrid multi-objective intervalvalued intelligent methods by merging multiple thoughts of different optimizers and designing some new operators in multi-objective intervalvalued environments, which is an open topic.
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