An effective model of multiple multi-objective evolutionary algorithms with the assistance of regional multi-objective evolutionary algorithms: VIPMOEAs

Applied Soft Computing 13 (2013) 2863–2895

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

An effective model of multiple multi-objective evolutionary algorithms with the assistance of regional multi-objective evolutionary algorithms: VIPMOEAs Hossein Rajabalipour Cheshmehgaz∗ , Mohamad Ishak Desa, Antoni Wibowo Faculty of Computer Science and Information Systems, Universiti Teknologi Malaysia (UTM), 81310, Skudai, JB, Malaysia

a r t i c l e

i n f o

Article history: Received 24 May 2011 Received in revised form 15 February 2012 Accepted 28 April 2012 Available online 26 May 2012 Keywords: Multi-objective optimization problems Multi-objective evolutionary algorithms Multiple MOEAs Island model Pareto Fronts

a b s t r a c t Division of the evolutionary search among multiple multi-objective evolutionary algorithms (MOEAs) is a recent advantage in MOEAs design, particularly in effective parallel and distributed MOEAs. However, most these algorithms rely on such a central (re) division that affects the algorithms’ efficiency. This paper first proposes a local MOEA that searches on a particular region of objective space with its novel evolutionary selections. It effectively searches for Pareto Fronts (PFs) inside the given polar-based region, while nearby the region is also explored, intelligently. The algorithm is deliberately designed to adjust its search direction to outside the region – but nearby – in the case of a region with no Pareto Front. With this contribution, a novel island model is proposed to run multiple forms of the local MOEA to improve a conventional MOEA (e.g. NSGA-II or MOEA/D) running along – in another island. To dividing the search, a new division technique is designed to give particular regions of objective space to the local MOEAs, frequently and effectively. Meanwhile, the islands benefit from a sophisticated immigration strategy without any central (re) collection, (re) division and (re) distribution acts. Results of three experiments have confirmed that the proposed island model mostly outperforms to the clustering MOEAs with similar division technique and similar island models on DTLZs. The model is also used and evaluated on a real-world combinational problem, flexible logistic network design problem. The model definitely outperforms to a similar island model with conventional MOEA (NSGA-II) used in each island. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Multi-objective optimization problems (MOPs) mostly deal with conflicting objectives and multiple optimal solutions. Their optimal solutions are known as Pareto solutions (PSs), – and its corresponding point in objective space, called Pareto Fronts (PFs) – which cannot be easily found and compared to other solutions [1]. Evolutionary algorithms (EAs) have been recognized as common techniques to optimize MOPs. EAs stochastically start to search with the aid of a set of (randomized) solutions, called initial population. The algorithms improve their population, iteratively, by some evolutionary operations and try to reach optimal solutions as close as possible. The main advantage of EAs, when applied to solve MOPs is the fact that they typically optimize sets of solutions placed in population. These EAs that deal with MOPs are called multi-objective EAs (MOEAs) – also known as evolutionary multiobjective optimization algorithms (EMOAs). MOEAs improve their population toward all possible directions in where there are possibilities of having PSs or at least reaching near to PSs, approximating PSs (known PSs – or known PFs in objective space).

∗ Tel.: +60 129412961. E-mail addresses: [email protected] (H. Rajabalipour Cheshmehgaz), [email protected] (M. Ishak Desa), [email protected] (A. Wibowo). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.04.027

Despite the benefit of using MOEAs, most real-world optimization problems often handle a large number of decision variables, objective functions and/or complex search spaces. Lots of iterations (or generation of new population) must be completed in order to approximate PSs effectively [2]. With multi-processing systems in the form of parallel or distributed environments, the computing time resulted from the increased iterations has been successfully reduced [3]. However, how MOPs can be efficiently divided into the multiple independent unite is still a challengeable effort to the algorithm designers [4]. One effective way is to subdivide the current population into sub-populations or to make multiple populations, which are even effective in solving MOPs. All (sub) populations are supposed to being improved independently by their heterogeneous or homogenous MOEAs [4]. These multi-populating cannot be effective without paying for some available drawbacks, which may be realized by some obvious restrictions in evolutionary operations [5]. Some evolutionary operations like mate (individual) selections require information about the entire search space and populations; whereas this division might preclude the selection performing effective. To overcome the shortage, many island models with immigration strategies and local or guided search enhancements have been proposed [3,5–10]. However, these models strongly depend on such frequent re-division process (central recollection and re-distribution process of all solutions form/to (sub)

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populations). It has been considered as a main disadvantage and an overhead computing in distributed systems that desire a high level of parallel. In this paper, first, a novel local MOEA that locally searches on a particular region is introduced. In spite of similar algorithms that ignore possible solution outside the region [3,5–10], the proposed MOEA does not enforce any hard constrain on the region. Instead, the algorithm benefits from the solutions outside (but nearby the region) in its generations. Moreover, the multiples of these local MOEAs are designed to search individual regions and assist other independent conventional MOEAs. While there is no frequent collection, division and distributing of (sub) populations in the local MOEAs, the algorithms work on an island model and communicate to each other through a novel immigration strategy. The remaining publication is structured as follows: initially, the general steps in typical MOEAs are presented and the main goals of MOEAs are briefly explained in Section 2 and in the following section, different island models used in MOEAs are reviewed. Then, the new local MOEA with all its novelties, are introduced and explained in Section 4. The novel proposed island model is presented in Section 5, while the division and immigration techniques are described in its following subsections. Before going for experiments in Section 7. Section 6 introduces two common performance metrics to evaluate the proposed algorithms. After evaluation through three experiments and one real-world test case, the scalability of the proposed model is discussed in Section 8. Finally, the conclusion and some future works are stated in Section 9. 2. Multi-objective evolutionary algorithms MOEA initializes the evolutionary computing with a population of solutions that is often created randomly and the iterative steps are explained as follows:

Initial population a. b. c. d.

Mate selection (making mate pool) Doing crossover and mutation Environmental selection (elitism strategy) If stopping condition is not satisfied go to step (a).

End

Fig. 1 illustrates the steps through an illustrative example with 10 solutions in its population. In the first step of iterative part, step (a), mate selection selects parents based on their fitness values. Each MOEA might use particular strategy to select the parent. Selection by switching objectives, aggregation selection with parameter variation and Pareto-based selection have been some techniques to do the selections [11]. In the next step, step (b), offspring are created with crossover and mutation. And finally in step (c), some of the current solutions in hand are selected through environmental selection. The two primary goals in MOEAs are to find better solutions in lower generations: (1) Converging to the true PSs or PFs, rapidly, as the generation goes forward. (2) Covering a variant area of feasible objective space. With a simple minimizing case, Fig. 2 demonstrates the position of known-PFs in 2D-objective space that are found by two different MOEAs on the same problem. Referring to Fig. 2a, MOEA2 shows rapid coverage to the true PFs while all known-PFs gained

by MOEA1 are dominated by all known-PFs in MOEA2 ’s population. In Fig. 2b, the current known-PFs in MOEA2 ’s population give much diversity as compared with MOEA1 where the respect solutions are nearly concentrated. Therefore, any selection operations (mate and environmental) should be regarded to improve the coverage speed and the diversity, simultaneously. 3. Island model and multi-MOEA Island models, which work with multiple (sub) populations evolving by local MOEAs in relative isolation, are designed upon some logical unites called islands [6,9]. Islands contain a number of solutions (individuals of populations) and communicate to each other via logical geometric structures such as rings, meshes and any shape of linkages. These algorithms are called distributed or multiple (populations) MOEAs working based in island model [4]. Basis on the literature, three general island models have been used as follows [4]: (1) Homogeneous Islands searching for overall PSs (or PFs). (2) Heterogeneous Island search for overall PSs (or PFs). (3) Homo or Heterogeneous Islands search for local PSs (or PFs) in predefined regions. In the first and the second model, the MOEAs in islands, search the overall search space for PSs, while their populations benefit from some immigration strategies. The only different is to set equal or different evolutionary parameters (such as crossover and mutation rates) or the shape of evolutionary operations (such as crossover form, and selection strategy). Apparently, both island models are effective due to utilizing each island to search the overall search space. There is no doubt that well-designed local evolutionary strategies can improve MOEAs with local recombination operations and somehow local fitness definitions [12–14]. Meanwhile, some multipopulation MOEAs have been developed on the third model to search for local PSs or PFs more effectively (e.g. [7–9,12–16]). However, there is no solid improvement with these models, because the algorithms must carry on some overhead activities to manage all possible shortages that might be resulted by the isolation and localization. Some need to employ a frequent division (e.g. [7,9,12–14]) and/or a frequent collection and distribution by a Central Island (e.g. [8]) to correct the local search directions. Some other like the work of Deb et al. [16] can only be used for particular type of MOEA that has the convex shape of feasible objective space. In general, the algorithms working on the third island model benefit from two ‘divide and conquer’ approaches: explicit and implicit. In explicit approach, MOEAs explicitly divide the respect problem into the islands through splitting the population or objective space (e.g. [6,9,12]). Whereas, some other MOEAs with the third island model, use the particular fitness functions that implicitly direct the search in islands into the specified regions [15,16]. 4. Very important population multi-objective evolutionary algorithm (VIPMOEA) The proposing algorithm is designed to search on a particular region of objective space, called VIP region. With the given polar angles of the region, a novel order system is developed to assign a particular fitness to solutions in the current population. New mate and environmental selections are also developed in order to search the region locally and effectively. In the following sub-sections, the order system, mate and environmental selections are introduced and explained.

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F1

F1

Initial Population

F1

F1

F2

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F2

F2

F2

a) Mate Selection

b) Generation

Parent Offspring

c) Environmental Selection

Iteratively

Fig. 1. MOEA steps with an illustrative example.

F1

Known-PFs found by MOEA 1 Known-PFs found by MOEA 2

PFs

F2

a) Coverage to the PFs by two sets of known-PFs gained by two different MOEAs in the same iteration Known-PFs found by MOEA1 F1

Known-PFs found by MOEA 2 F1

F2

F2

b) Diversity gained by two different sets of known-PFs by two MOEAs Fig. 2. Two examples showing the quality of known-PFs.

Fig. 3. Ordering in VIPMOEA regarding the assigned region.

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Fig. 4. Selecting three solutions regarding to Crowding Distance technique.

4.1. VIP order systems In this system, three groups of solutions (individuals in current population) are defined. First group includes all the first fronts (known-PFs) inside the region (black colored solutions in Fig. 3), and the rest except dominated solutions (means that there is at least one solution that all its objective values are less or equal to this solution) outside the region are placed in the second group (straightline and gray colored ones Fig. 3). And the last group consists of all dominated solutions outside the region (the white ones in Fig. 3). Initially, the order numbers are assigned to the groups in order to all members in a lower group get the lower numbers related to the members of the groups with higher numbers. Afterward, inside the groups, each solution is assigned an order number as it is explained in the following steps. 4.1.1. Local ordering in the first group All solutions in the first group, gain high priorities to be selected, but these solutions still require a local ordering system. Because, it might be a case that the number of solutions in the first front, are more than the number environmental selection requires. In Fig. 4, nine solutions, including six solutions and three solutions in the first and the second fronts respectively, are inside the VIP region. Suppose that the environmental selection requires only three solutions from the first front. In this situation, the ordering systems with the concept of crowing distance [17], is used and the solutions with low levels of crowding area nearby are assigned lower order numbers as it is illustrated in the figure. 4.1.2. Local ordering in the second group To order the solutions of this group, first, an ideal point is calculated with the current extreme points inside the VIP region. In minimizing case, the point can be calculated with minimum values of all current objective values gained by all the solutions inside the region. This point is recognized as a base to order the solutions. Each solution in this group that has fewer Euclidean distances will gain a lower number. In Fig. 3, the ideal point is marked by a square shape. In some case, VIP region might cover the infeasible region of the respect MOP. To order the solutions, the ideal point is first calculated with all solutions without considering the VIP region. And then the point is rotated into the region based on the zero point and into the mid-angles of the VIP region as it shows in Fig. 5. 4.1.3. Local ordering in the third group The rest of solutions that are placed into the last group, third group, are also treated like the second group. With this ordering system, each solution has an order number regarding two qualities: first, being known-PF inside VIP region,

Fig. 5. Ordering solutions in objective space based on the rotated ideal point.

and if not, how far it is close to the ideal point (or rotated ideal point). 4.2. Mate selection As a first and important evolutionary operation, mate selection works with an ordering system. A binary tournament selection approach is used to design the mate selection here as follows. Different situations are considered, and the decision in selection is made regarding to the VIP region. Mate-Selection (Current Population, Angle) Randomly, select two different solutions, A and B from Current Population If (A and B are inside VIP region) ---------------------------If Order(A) < Order(B) A is selected; Else If Order(A) > Order(B) B is selected; Else Randomly, A or B is selected; End

A B

End End

---------------------------

If (A and B are outside VIP region) -------------------------If Order(A) < Order(B) A A is selected; B Else If Order(A) > Order(B) B is selected; Else If EuclideanDistance(A, Ideal Point) < EuclideanDistance(B, Ideal Point) A is selected; Else B is selected; End End End End -------------------------If (A is Inside VIP region) && (B is Outside VIP region) A is selected; End If (A is Outside VIP region) && (B is Inside VIP region) B is selected; End

A B

End

4.3. VIP environmental selection (VIP-ES) The environmental selection is conducted to create a new population from current population and new offspring. In each generation, the offspring generated from the respect parents can be

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Fig. 6. VIP environmental selection with different number of selected solutions.

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Fig. 7. Known-PFs in the current populations of VIPMOEA and NSGA-II by 100 iterations.

Fig. 8. Known-PFs in the current populations of VIPMOEA and NSGA-II by 600 iterations.

involved in creating the next population through two main strategies: ( + )-ES and (, )-ES. The first strategy,  solutions are selected from the union set of parents and offspring, whereas in the second strategy, only the  solutions form  offspring are chosen as new population for next generations. We used the first strategy to develop the proposing environmental selection, VIP environmental selection (VIP-ES) to create the next population. The selection works based on VIP ordering system, and the solutions can be selected in next population regarding the three groups in VIP ordering system and the local order number in each group. VIP-ES is explained through an illustrative example as follows. Fig. 6 shows 10 solutions, including five parents and five offspring, and VIP-ES tries to select some of these solutions for next generations. Simply, the figure shows how VIP-ES selects three (Fig. 6c), four (Fig. 6d), five (Fig. 6e) and six (Fig. 6f) solutions and the selected solutions are identified by the closed line in each figure.

are executed for 100 iterations on DTLZ1. Known-PFs in their final populations are presented in Fig. 7. Although the VIP region covers a region with PFs inside, some known-PFs gained by VIP-MOEA, are placed outside the regions (see Fig. 7). However, all knownPFs in VIPMOEA’s population are laid inside as the generation goes forward (by 600 iterations) (see Fig. 8). In the second test with DTLZ5, we intentionally assign a VIP region to VIPMOEA to cover the part of objective space where no PFs is located in (see ˛ in Fig. 9). Even with the VIP regions covering the worthless area, VIPMOEA is able to find some known-PFs outside, but nearby – see Fig. 9a (comparing with the conventional MOEA, NSGA-II). While the generation continues, VIPMOEA finds the known-PFS that are closer to the VIP region as Fig. 9b shows it.

5. Improving MOEAs with multiple VIPs 4.4. Testing VIPMOEA To show how the algorithm works, we run two VIPMOEAs with their particular VIP regions on DTLZ1 and DTLZ5. In the first test case, the VIP region is intentionally set to cover a region in objective space including some PFs whereas in DTLZ5, the angles of VIP region is set to covers an area with no PFs inside. In the first test, the VIP region is identified with the angles of ˛ (see Fig. 7), and the algorithm with a conventional MOEA, NSGA-II,

Many island models with homogeneous or heterogeneous MOEAs in their islands have been developed to benefit from multiple MOEAs to find better solutions in terms of coverage speed and diversity. In this research, the advantage of VIPMOEAs is considered to help other conventional MOEAs through an island model with a sophisticated immigration strategy. Fig. 10 illustrates the proposing island model with one island, called Central Island, where the conventional MOEA works. The model also includes

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Fig. 9. Known-PFs in the current populations of VIPMOEA and NSGA-II by 600 iterations.

Central Island with MOEA (e.g. NSGA-II or MOEA/D)

Immigration Strategy

VIP Island #1

Local Population Local VIP-MOEA

Local Population Local VIP-MOEA

Local Population Local VIP-MOEA

VIP Island #2

VIP Island #3

Local Population Local VIP-MOEA

VIP Island #n

Fig. 10. The island model used to improve MOEA in Central Island with VIP Islands.

multiple VIP Islands where local VIPMOEAs individually search on their VIP regions and supply Central Island with their good solutions. This model with multiple algorithms, including a conventional MOEA and multiple VIPMOEAs manipulating each other, works such a meta-algorithm that can be described as follows: Step 1: Space division

In spite of the previous division techniques like clustering [18] and the polar-base division by Sato et al. [12] that explicitly split their population, the proposing division technique divides the objective space. The MOEA, in Central Island, divides the objective space with the known-PFs that have been recently updated in its current population. Afterward, the divisions are assigned to the VIP Islands in order to search the regions locally and individually.

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(2) Transferring the solutions {xi }1≤i≤ord(X) (∈ X) to polar coordiStep 1: Dividing the space by MOEA. Step 2: Assigning the angles (VIP regions) to the VIPMOEAs. Step 3: Local evolutionary computing (MOEA and Its VIPMOEAs). Step 4: Immigration form VIPs to MOEA’s population. Step 5: Adjusting VIPs’ Angles based on new population in MOEA. Step 6: Immigration from MOEA to VIPMOEA’s populations. Step 7: If stop conditions is not satisfied go to step 3.

i ), ∀ 1 ≤ i ≤ ord(X) nates in the form of xi = (r i , 1i , 2i , . . . , m−1 (3) Clustering and dividing by the angles according to Eq. (2), as shown with an illustrative example in the next.

Illustrative example: Let m = 3, d2 = 4

            1 2 3

Pop =

4 1

3 2

,

2 1

,

,

4 1 1

,

3 3 2

,

4 3 2

        Although the MOEA splits the space with the help of current knownPFs in its population, VIP Islands have own independent populations evolving with their local evolutionary operations designed for the respect region (VIP mate and VIP environmental selections). However, the direction of search in each island is defined and iteratively adjusted by the conventional MOEA in Central Island. The proposing division technique works based on the polar coordinate system. The first utilization of polar system backs to the works of Bian et al. [19] and Sierra and Echeverria [18] that have proposed genotypes that are transferred to polar coordinates called “polar genotype.” According to their work, the corresponding polar coordinate of an n-dimensional Cartesian coordinate is given as (r, 1 , 2 , . . . , n−1 )

(1)

where 0 ≤ n−1 ≤ 2, 0 ≤ q ≤  for q = 1, . . . , n − 2 and r > 0. And each solution (individual), X in population can be transfer to Cartesian coordinates, X = [x1 , x2 , . . . xn ] such that:

1 4 1

Known − PFs =

2 3 2

3 2 1

4 1 1

⎧⎡ ⎤⎫      4.2 4.1 3.7 4.2 ⎨ r ⎬ 1.3 0.9 0.5 0.2 X = ⎣ 1 ⎦ ∼ = ⎭ ⎩ 0.2 0.5 0.2 0.2 2

• Clustering X based on 1 :  = {1 }



A1 =

0

[Center1 , Center2 , . . . , Centerd−1 ] = k − means(, d − 1)]

 /2

x1 = r cos n−1 sin 1 . . . sin n−2 x2 = r sin n−1 sin 1 . . . sin n−2 x3 = r cos 1 sin 2 . . . sin n−2



A1 = 0



... xi = r cos i−2 sin i−1 . . . sin

Example continued:

0

0.72

[[0.72] = k − means({1.3, 0.9, 0.5, 0.2}, 1)] /2



n−2

... xn = r cos n−2 In the proposing technique, having positive values for all objective functions is assumed and the union of all division must cover overall positive part of space. Let Pop: Population of solutions (individuals) of MOEA in the Central Island N: Pop’s size m: Dimension of objective space dm−1 : Number of divisions [Angles2×(m−1) ]dm−1 : (Declination) angles of divisions

• Dividing X based on A1 : A1 (a) = {2 |A1 (a) ≤ 1 < A1 (a + 1)},

∀ 1 ≤ a ≤ d,

T

∀ (r, 1 , 2 ) ∈ X

Example continued: A1 (1) = {0.2, 0.5} A1 (2) = {0.9, 1.3}

Fig. 11 shows 2D and 3D-objective space with four divisions and the respect angles. The steps of the division technique are presented as follows: (1) X = {all known − PFs of Pop} While ord(X) < dm−1 // the number of X’s members { y ∈ Non-dominated (Pop − X) X = X ∪ {y} }

• Clustering A (a) (∀1 ≤ a ≤ d) based on 2 : 1

 A2,a =

0

[Center1 , Center2 , . . . , Centerd−1 ] = k − means(A1 (a) , d − 1)]

 /2



/2 =

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Example continued:



A2,1 = 0 [[0.35] = k − means({0.2, 0.5}, 1)]   = 0 0.35 /2

/2

A2,2 = 0 [[0.11] = k − means({0.9, 1.3}, 1)]   = 0 0.11 /2

/2



Example continued:





Angles1 =



 Angles2 =

 Angles3 = ◦ In the case of more dimensions (m > 3), the clustering and dividing must be continuous as follows:

⎧ | z ⎪ ⎪ ⎪ ⎪ A ⎨ 1 (n1 ) ≤ 1 < A1 (n1 + 1), An

z−1 ,nz−2 ,n1

(a)

=

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

(n2 ) ≤ 2 < A2,n1 (n2 + 1)

A

2,n1 ⎪ ⎪ ⎪ ⎪ ⎩ ,...,

⎪ ⎪ ⎪ ⎪ ⎭

Anz−1 ,...,n1 (a) ≤ z−1 < Anz−1 ,...,n1 (a + 1) ∀ 1 ≤ a ≤ d, ∀ (r, 1 , 2 , . . . , m−1 )T ∈ X

∀ 1 ≤ z < m, 1 ≤ nz ≤ d,

 Anm−1 ,nm−2 ,...,n1 ,a =

 0

k − means(An

∀ 1 ≤ i < m, 1 ≤ ni ≤ d, 1 ≤ a ≤ d

m−2 ,nm−3 ,...,n1

(a) , d





[Center1 , Center2 , .., Centerd−1 ]

/2

− 1)

Example continued: A2,1 (1)={0.2} , A2,1 (2)={0.5} , A2,2 (1)={0.9} , A2,2 (2)={1.3}

• Setting final angles of all divisions as follows:

 Angles1 =

 Angles2 = ...



Anglesd =

A1 (1)

A2,1 (1)

...

A1 (2)

A2,1 (2)

A1 (1)

A2,1 (1)

A1 (2)

A(d−1),1,...,1 (1)

...

A(d−2),1,...,1 (2)

A(d−1),1,...,1 (2)

...

A(d−2),1,...,1 (1)

A(d−1),1,...,1 (2)

A2,1 (2)

...

A(d−2),1,...,1 (2)

A(d−1),1,...,1 (3)

A1 (1)

A2,1 (1)

...

A(d−2),1,...,1 (1)

A(d−1),1,...,1 (d)

A (2)

A2,1 (2)

...

A(d−2),1,...,1 (2)

A(d−1),1,...,1 (d + 1)

1 Anglesd+1 = ... Angles2d = ...



A(d−2),1,...,1 (1)





A1 (1)

A2,1 (1)

...

A(d−2),1,...,1 (2)

A(d−1),1,...,1 (1)

A1 (2)

A2,1 (2)

...

A(d−2),1,...,1 (3)

A(d−1),1,...,1 (2)

 

A1 (1)

A2,1 (1)

...

A(d−2),1,...,1 (2)

A(d−1),1,...,1 (d)

A1 (2)

A2,1 (2)

...

A(d−2),1,...,1 (3)

A(d−1),1,...,1 (d + 1)

⎡ ⎢

Anglesdm−1 = ⎣

A1 (d) A1 (d + 1)



A2,(d−1) (d)

...

A(d−2),(d−1),...,(d−1) (d)

A(d−1),(d−1),...,(d−1) (d)

A2,(d−1) (d + 1) A(d−2),(d−1),...,(d−1) (d + 1)

...

⎤ ⎥ ⎦

A(d−1),(d−1),...,(d−1) (d + 1)

(2) Step 2: In this step, the regions created by the angles are assigned to VIPMOEAs to search on the region locally and individually. Although the regions cause more interest of solutions inside the regions rather than outside, VIPMOEAs do not completely ignore the solutions outside the respect regions. Therefore, the algorithms

 Angles4 =

A1 (1)

A2,1 (1)

A1 (2)

A2,1 (2)

A1 (1)

A2,1 (2)

A1 (2)

A2,1 (3)

A1 (2)

A2,2 (1)

A1 (3)

A2,2 (2)

A1 (2)

A2,2 (2)

A1 (3)

A2,2 (3)

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 =



 

=



 =





0 0 0.75 0.35 0 0.75

0.35 /2

0.75 /2

0 0.35

0.75

0.35

/2

/2

   

=

can benefits from other good solutions even outside, but nearby, whereas previous techniques disregard these solutions. In Fig. 12, steps 1 and 2 of a problem with two objectives and four VIP Islands (four divisions) are illustrated. The centers of K-mean specify the borders of VIP regions as these are calculated with the division technique aforementioned. Step 3: All islands, including central and VIPs; evolve their populations with their local evolutionary operations. In Central Island, MOEA can be a conventional MOEA like NSGA-II [17], MOEA/D [9], SPEA2 [20] or other types of MOEAs. All VIP Islands have their own VIPMOEAs with their regions that are assigned by Central Island. VIPMOEAs benefit from the mate and environmental selection operations that designed for their VIP regions and introduced earlier. Step 4: Iteratively, after one (or more) generation in VIP Islands, the islands try to send a copy of a predefined number of its solutions (as emigrants) to Central Island. Each VIP Island, first, tries to send the known-PFs which are located in the VIP region, if there is; otherwise it selects some other known-PFs outside the regions. Fig. 13 illustrates step 4 in detail with an example that has been already introduced in Fig. 12 with only one emigrant for each VIP Island. It might happen that some VIP Islands do not find any knownPF inside the assigned region. In this case, the island sends a copy of other known-PFs instead. In Fig. 13, VIPk sends another knownPF outside its VIP region. This VIP is called as Barren Island for the current iteration. With all new immigrants in Central Island; the island receives more solutions than its population size that must be truncated back to the population size. Therefore, an environmental selection is required to truncate these solutions (including solutions in population and new immigrants). We use Crowding Distance technique [12] for the environmental selection. Fig. 14 shows the selection with declining three solutions shown in the figure. Step 5: Similar to steps 1 and 2, Central Island divides the objective space again with the help of new information of knownPFs (updated by the recent immigrants). The updated angles with respect VIP regions are sent to the related VIP Islands to change their search direction, if it needs (see Fig. 15 illustrating the step). Step 6: It is obvious that migration efforts between any two MOEA’s populations can improve them in next generations. However, with the change of direction of search in VIP Islands, it might cause some VIPMOEAs finding their population with no solution inside the new assigned regions. Therefore, a guided migration strategy is required to preclude the situation that no VIP region (in VIP Islands) remains empty. Otherwise, it might face overhead in computing and being barren (means that the island does not contribute in exploring the assigned region).

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The guided migration strategy is considered to select a predefined number of known-PFs in Central Island that are located in the VIP regions and sending them to the respect VIP Islands. These

emigrants will be added to the VIPMOEA’s populations as it shows in Fig. 16. The VIP Islands with new immigrants also need to truncate the current solutions, including current population and new immigrants. As VIP Islands are more interested in the solutions inside their assigned VIP regions, then VIP environmental selection is used. Fig. 17 shows the selection through the illustrative example.

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MOEA's Population and immigrants in Iteration t F1

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F2 VIPMOEA j th Population in Iteration t Fig. 15. Step 5: adjusting VIP regions’ Angles in VIP Islands based on changes in Central Island’s population.

6. The performance metrics Due to the goals in MOEAs discussed in earlier section, multiple performance metrics should be used for comparison the performances of different MOEAs. In the experiments here, the following metrics are used.

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percentage of the solutions in B that are dominated by at least one solutions in A and it is formulated in Eq. (3). CR(A, B) =

|{b ∈ B|∃a ∈ A : a DOMINATES |B|

b}|

(3)

/ 1 − CR(B, A). where it is not necessarily that CR(A, B) = 6.2. Hypervolume (HV)

6.1. Coverage rate (CR) Supposing that there are two sets of known-PFs, A and B, in the final populations of two different MOEAs. CR(A, B) is defined as the

This volume is used to shows what volume of objective space is enclosed by the solutions of A and a predefined point as reference point.

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Fig. 18 shows how CR and HV can be calculated with two sets of solutions: black and white points in a 2D-objective space.

7. Results The results are presented in two individual subsections. First subsection, Section 7.1, is concerned with conventional experiments and the related results on DTLZ test cases. We evaluated all algorithms designed and presented in this research through some theoretical experiments. In the second part, a real-world combinational optimization problem, 4-level flexible logistic network design with three conflict objectives of time, facility cost and transportation cost is briefly introduced. The objectives are considered to be optimized simultaneously; therefore, the proposed algorithms are used and compared with the conventional MOEAs through the problem optimization.

7.1. Theoretical experiments result Three experiments are conducted on a variety of DTLZ test cases: DTLZ1, DTLZ2, DTLZ3, DTLZ5 and DTLZ7. All algorithms are individually run on a computer with a CPU of Intel Core Due 1.87 GHz with no parallelization technique. We use independent sampling during 35 independent runs. In the first experiment, the proposed island model with MOEA and VIPMOEAs (called VIP-MOEA afterward), is compared with the corresponding MOEA using clustering technique and proposed by Streichert’s work [9]. He performance of the division techniques used in both algorithms are evaluated and compared on the same number of divisions. Each division in Clustering MOEA (called CMOEA afterward) is considered as one island using NSGA-II whereas in VIP-MOEA, only Central Island runs NSGA-II, and all VIP Islands use their own VIPMOEA. The two introduced performance metrics: hypervolume, coverage rate along with real CPU-time, are considered to be calculated and used to compare the algorithms’

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895 VIP-MOEA

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c) 7.1.2. Comparison result As it mentioned, both algorithms are executed for 35 individual runs and 600 generations pre run while 16 clusters (or VIP regions) are given to the algorithms. In each 100 iterations (generations), the

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7.1.1. Experiment 1: VIP-MOEA vs. CMOEA In addition to the aforementioned conditions, both algorithms must use the same number of individuals (solutions) in each generation. Therefore, the size of main population in CMOEA is equal to the size of the union of all islands’ populations in VIP-MOEA (Central and VIP Islands). The number of VIP regions (or VIP Islands) is equal to the number of clusters in CMOEA. Additionally, Fig. 19 illustrates the most dissimilarity between two algorithms. CMOEA iteratively performs (re) collection and (re) distribution of sub-populations after clustering in the main population whereas VIP-MOEA benefits from its migration strategy. CMOEA exchanges a large number of individuals in each (re) collection and (re) distribution acts while VIP-MOEA consistently exchanges only a low-number of individuals (migrants). To compel the dissimilarity during the computing, the number of moveable individuals should be maintained in the same for both algorithms (either copying or moving). For instance, consider a case of 16 clusters (or VIP regions) and 370 solutions in CMOEA’s population. We need to keep the immigration in each single generation of VIP-MOEA, if CMOEA performs re-collection and re-distribution after each 10 iterations. Actually, it guaranties the same number of solutions that moved in both algorithms (see Table 1). Some other settings that make the condition fair are explained in Table 1.

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performances. Other factors and conditions that might influence the performance are set and discussed in the related and following sub-sections. In the next two experiments, VIP-MOEA is individually compared with two popular MOEAs: NSGA-II [17] and MOEA/D [21]. NSGA-II is Pareto-based algorithm, and all its evolutionary strategies are designed based on this concept, whereas the MOEA/D uses the concept of decompositions and sub-problem with scalar optimization techniques. These two different MOEAs are selected to be improved with VIPMOEAs here. However, due to using multiple populations in VIP-MOEA, we use the same number of populations and a similar island model with the same form of linkage, immigration rate and the number of migrants for these experiments. It helps to make the equal experiment conditions for the algorithms. The performance metrics: hypervolume and coverage rate are also considered to be calculated and evaluated. The details and the initial setting are given in the following sub-section. As the evolutionary operations significantly influence MOEA, the same crossover and mutation operations are used. We use discrete two-point-crossover, and the mutation and the crossover probabilities of 0.1 and 0.95, respectively, for all implementations of MOEAs (including VIP-MOEA, multi-NSGA-II and multi-MOEA/D). The number of migrants is also fixed to 2 solutions in all circumstances.

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Table 1 Parameter setting for the experiment condition. Parameter

Number of clusters (or VIP region)

Size of main population in CMOEA

Sum of the size of all population in all islands

Number of migrants (in VIP-MOEA)

Frequent immigration in VIPMOEA

Frequent re-collection and re-distribution in CMOEA

Number of individuals moving in each 10 iterations

Value

16

370

Central Island: 50 VIPs: 20 Total: 50 + 16 * 20 = 370

2

Each iteration

Each 10 iterationsa

6400 for both

In these iterations, the number of individuals moved in CMOEA so far is equal to number of migrants in VIP-MOEA.

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7.1.3. Experiment 2: VIP-MOEA vs. multiple NSGA-IIs In addition to the setting discussed earlier, both algorithms here are run on a similar island model with the same size of population in the corresponding islands (see Fig. 26). As Fig. 26 shows, all islands in multiple NSGA-IIs (called GP afterward) use homogenous NSGAIIs. Its corresponding island with Central Island in VIP-MOEA has 50 solutions in its population, while the other islands evolve their populations with the size of 20. As the comparison must be performed in a fair condition; more settings for both algorithms are considered and explained in Table 2. VIP Islands are supposed to improve the conventional MOEA in Central Island; therefore, in addition to the overall result gained from all islands, the metrics (CR and HV) are also evaluated in Central Islands for both algorithms. Fig. 27 draws an interesting picture of how the both algorithms improve their populations in Central Islands only and the whole island models in iteration 300 on DTLZ1. Obviously, the number of known-PFs given by all islands in both algorithms is larger. However, their absolute quality in terms of higher values in CR and HV is not confirmed yet (it will be discussed in comparison result sub-section). Additionally, to see the difference in their coverage speed, Fig. 28 presents a picture of known-PFs found in early generations (120 iterations) and later generations (600 iterations) on DTLZ3. Clearly, as the generations continuous, the difference in the quality between

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values of three performance metrics, hypervolume (HV), coverage rate (CR) and CPU-time, are calculated and recorded. Comparing CMOEA and VIPMOEA on DTLZ1 definitely confirms the better performance with VIPMOEA. The interval boxes in Fig. 20, shows the 95% confidence interval (CI) of these parameters estimated in the iteration of 100, 200, 300, 400, 500 and finally 600. Referring to CR values, VIP-MOEA is significantly better than CMOEA, particularly in the early generations. Although, there is no significant similar advantage according to HV in this case, VIPMOEA shows higher values of HV in early generations and keeps the level as the generation goes forward. Meanwhile, the real CPUtime spent by CMOEA is significantly higher than the corresponding time in VIP-MOEA (around 1200 s more). It shows that CMOEA is significantly slow as compared with VIP-MOEA. The comparison result for DTLZ2 and DTLZ5 is somehow disappointing, see Figs. 21 and 24. CR and HV gained by VIP-MOEA are noticeably low as compared with the related values in CMOEA. However, the real CPU-time spent by CMOEA is still higher, and it is very slow in computing as compared with the consuming time in VIP-MOEA (see Fig. 21c – the difference is about 800 s, see Fig. 24c – the difference is about 900 s). The same improvements are gained on the rest of DTLZ test cases with VIP-MOEA. DTLZ3, DTLZ4 and DTLZ7 give somehow the higher values in CR and HC and lower time in CPU-time with VIPMOEA as the respect results given by CMOEA are compared, see Figs. 22, 23 and 25.

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Fig. 23. Comparison results of CMOEA and VIP-MOEA on DTLZ4.

the two sets of known-PFs is declined. However, it is important that we identify a better algorithm that keeps its gained quality at least with not giving the values of CR of HV less than others as the generation goes forward.

analyzed in the iteration of 100, 200, 300, 400, 500 and finally 600. According to CR and HV-gained during the computing, VIP shows better improvement not only in Central Island but also in the whole island model. Similar to the previous experiment, the respect result in test case of DTLZ2 is still disappointing on HV value (see Fig. 30). HV gained by VIP are noticeably low comparing with the related values given by GP in different generations. Besides, VIP gives the higher values in CR in the early generations (about 300 generations) however, the difference is nearly declined in the later generations (in iteration 600). The related result in the test cases of DTLZ3 and DTLZ4 shows the significant improvement on CR metric given by VIP in not only Central Island but also in all islands (see Figs. 31 and 32) in all islands

7.1.4. Comparison results Both algorithms, GP and VIP, are evaluated with 35 individual runs and 600 generations per run while 16 clusters and VIP regions are given to the algorithms. In each 100 iterations (generations), the values of two performance metrics, CR and HV gained in their Central Islands, and all islands are calculated and recorded. Comparing GP and VIP on DTLZ1 confirms the better performance with VIP in both, Central Island and all islands. The interval boxes in Fig. 29 shows the 95% confidence interval of the metrics

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Fig. 24. Comparison results of CMOEA and VIP-MOEA on DTLZ5.

Fig. 25. Comparison results of CMOEA and VIP-MOEA on DTLZ7.

are slightly lower than the respect values in GP. However, both algorithms reach nearly the same values of HV as the generation continuous (after around 300 generations). In the test case of DTLZ5, the performance metrics, CR and HV are noticeably improved with VIP in Central Island. However, in the whole islands, GP considerably gives the better results in CR and HV as compared with the corresponding CR and HV in VIP. In DTLZ7, VIP gives higher values of CR and HV in Central Island and all islands. Although, VIP and GP nearly reach the same values for CR and HV after almost 500 generations, the coverage speed (recognized by CR) and the diversity among known-PFs (recognized by HV) in the earlier generations (about 300 generations) for both algorithms are noticeable.

7.1.5. Experiment 3: VIP-MOEA vs. multiple MOEA/Ds In this experiment, we use a recent MOEA, MOEA/D [21] instead of NSGA-II in the previous experiment. With no change, all conditions in Experiment 2 are considered again here. With the comparison of the multiple MOEA/Ds (called GP afterward) and MOEA/D&VIPs (called VIP afterward) on DTLZ1, the result shows the better performance with VIP (see Fig. 35). HV values gained by VIP (see Figs. 33 and 34). The values of CR and HV gained by VIP are generally higher than the respect values in GP, particularly when the generation continuous. The similar results are acquired by the algorithms on DTLZ2, DTLZ3, DTLZ4 and DTLZ7; see Figs. 36–40. Unfortunately, the result in Central Island in DTLZ5 is still disappointing; however, the CR

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Multiple NSGA-IIs

VIP-MOEA NSGA-II

NSGA-II

Migration VIP1

2879

Migration VIP2

VIPn

NSGA-II 1

NSGA-II 2

NSGA-IIn

Fig. 26. Two Island models with the same structure and the related MOEA in each island.

Table 2 Parameter setting for the experiment. Parameter

Number of Central Island

Size of the population in Central Island

Number of non-Central Island (VIP Islands for VIP-MOEA)

Size of the population in non-Central Island

Frequent immigration

Number of migrants

Value

1

50

16

20

Each iteration

2

Fig. 27. Known-PFs in the Central Island’s population (left) and in the whole of model (right) in iteration 300 on DTLZ1.

Fig. 28. Known-PFs the whole of model after 120 and 600 iterations on DTLZ3.

2880

Ref. Point=[500 500 500] 95% CI for the Mean of HV in Centeral Island's Populations 125000000

0.8

124980000

Hypervolume

1.0

0.6

0.4

0.2

124960000

124940000

124920000

124900000

0.0

) GP 0(

10

P)

0

10

I (V

) ) ) ) ) P) P) P) P) P) GP GP GP GP GP VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 20 30 40 50 60 20 30 50 40 60

0 10

P) P) P) P) P) P) P) P) P) P) P) P) VI VI VI (G (G VI 0 (G VI 0 ( G (G VI 0 (G 0 ( 300 0 ( 500 0( 0( 0( 0( 40 20 60 20 10 40 50 30 60

Generation

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ1

95% CI for the Mean of CR in the all Populations

Ref. Point=[500 500 500] 95% CI for the Mean of HV in all Populations

1.0

125000000 124999000 124998000

Hypervolume

Coverage Rate

0.8

0.6

0.4

124997000 124996000 124995000 124994000 124993000

0.2

124992000 124991000

0.0 10

) GP 0(

) VIP 0(

10

) ) ) ) ) P) P) P) P) P) GP GP GP GP GP VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 30 20 50 60 40 30 20 40 50 60

Generation

10

) ) ) ) ) ) ) ) P) P) P) P) IP IP IP GP G P (VI GP GP GP VI VI 0 ( 00 (V 00 ( 00 ( 0 (G 0 (V 00 ( 0 ( 00 (V 00 ( 00 ( 0 2 4 30 6 50 30 2 1 40 5 6

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ1 Fig. 29. Experiment result of running GP (multiple NSGA-IIs) and VIP (VIP-MOEA) on DTLZ1.

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

95% CI for the Mean of CR in Center Island's Populations

95% CI for the Mean of HV in Centeral Island's Populations

0.8

124999500

Hypervolume

125000000

0.6

0.4

124999000

124998500 0.2 124998000 0.0

) GP

0(

10

) VIP

0(

10

0( 20

GP

) 0(

20

VI

P)

) GP

0(

30

P)

VI

0( 30

) GP

0( 40

P)

VI

0(

40

0(

50

) GP

0(

50

VI

P)

) P) GP VI 0( 0( 60 60

) ) ) ) ) ) ) ) ) P) P) P) IP IP IP GP GP GP (VI GP GP GP VI VI 0 ( 00 (V 00 ( 0( 00 (V 00 ( 00 (V 0 ( 00 ( 0 0( 20 50 30 6 4 10 3 40 2 5 6

0(

10

Generation

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ2 95% CI for the Mean of HV in all Populations 125000000

0.8

124999500

Hypervolume

Coverage Rate

95% CI for the Mean of CR in the all Populations 1.0

0.6 0.4

124999000

124998500 0.2 124998000

0.0 0 10

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

95% CI for the Mean of CR in Center Island's Populations 1.0

(

) GP

0 10

(V

IP)

0( 20

G

P) 20

VI 0(

P)

0( 30

G

P) 0 30

(V

IP)

0( 40

) GP

VI

0(

40

Generation

P)

0( 50

G

P) 0 50

I (V

P)

) P) GP VI 0( 0( 60 60

0( 10

) ) ) ) ) ) ) ) ) P) P) P) IP IP GP (VIP GP GP GP GP GP VI VI VI 0 ( 00 ( 0 ( 00 (V 00 ( 00 ( 0 ( 00 (V 00 ( 00 ( 0 6 20 50 4 30 4 6 3 2 5 10

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ2 Fig. 30. Experiment result of running GP (multiple NSGA-IIs) and VIP (VIP-MOEA) on DTLZ2.

2881

2882

95% CI for the Mean of CR in Center Island's Populations

95% CI for the Mean of HV in Centeral Island's Populations

1.0

125000000

Hypervolume

124900000 0.6 0.4

124800000

124700000

0.2

124600000

0.0

) ) ) ) ) ) P) P) P) P) P) P) GP GP GP GP GP GP VI VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 60 40 20 50 30 10 60 40 30 50 20 10

) ) ) ) ) ) ) ) ) P) IP) IP IP IP) IP GP GP GP GP GP GP VI 0 ( 0 0 (V 00 ( 00 (V 00 ( 00 ( V 00 ( 00 (V 0 ( 0 0 (V 00 ( 00 ( 30 6 2 5 4 6 1 5 2 3 4

10

Generation

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ3 95% CI for the Mean of HV in all Populations 125000000

0.8

124980000

Hypervolume

Coverage Rate

95% CI for the Mean of CR in the all Populations 1.0

0.6 0.4

124960000 124940000 124920000 124900000

0.2

124880000 0.0

) GP

0(

10

0( 10

P) VI

0(

20

G

P) 0 20

(V

I

P)

) ) ) ) P) P) P) P) GP GP GP GP VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 40 60 30 50 40 30 50 60

Generation

) ) ) ) ) ) ) ) ) P) IP) IP) IP IP IP GP GP GP GP GP GP (VI 0 ( 00 (V 00 ( 00 (V 00 ( 0 0 ( V 00 ( 00 (V 0 ( 0 0 (V 00 ( 0 0 0 4 5 2 6 3 10 6 3 2 5 4 1

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ3 Fig. 31. Experiment result of running GP (multiple NSGA-IIs) and VIP (VIP-MOEA) on DTLZ3.

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

0.8

95% CI for the Mean of HV in Centeral Island's Populations

0.8

124900000

Hypervolume

125000000

0.6 0.4

124800000

124700000 124600000

0.2

124500000 0.0 0( 10

) GP

IP)

V

0( 10

0(

G

P)

20

20

0(

VI

P) 0

30

P) (G

IP)

V

0( 30

0( 40

) GP

VI

0( 40

P)

0(

50

G

P)

0( 50

VI

P)

0( 60

G

P)

IP)

V

0( 60

10

0(

GP

) 0 10

(V

IP

) 0 20

(G

P)

0( 20

) ) ) ) ) ) P) IP) IP) IP IP GP GP GP GP VI 0( 00 (V 00 ( 00 (V 00 ( 0 0 ( V 00 ( 00 (V 5 4 30 6 4 5 6 3

Generation

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ4

95% CI for the Mean of HV in all Populations 125000000

0.8

124999800

Hypervolume

Coverage Rate

95% CI for the Mean of CR in the all Populations 1.0

0.6 0.4

124999600 124999400 124999200

0.2

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

95% CI for the Mean of CR in Center Island's Populations 1.0

124999000 0.0

) GP

0(

10

0( 10

P) VI

0(

20

GP

)

IP

0 20

(V

) 0 30

P) (G

) GP

P)

VI

0( 30

0(

40

P) VI

0( 40

Generation

0 50

P) (G

0( 50

P) VI

60

0(

) GP

0( 60

P) VI

) ) ) ) ) ) ) ) ) P) IP) IP) IP IP IP GP GP GP GP GP G P (VI 0 ( 00 (V 00 ( 00 (V 00 ( 0 0 ( V 00 ( 00 (V 0 ( 0 0 ( V 00 ( 0 0 0 6 4 3 2 10 5 5 6 4 2 3 1

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ4 Fig. 32. Experiment result of running GP (multiple NSGA-IIs) and VIP (VIP-MOEA) on DTLZ4.

2883

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95% CI for the Mean of CR in Center Island's Populations

95% CI for the Mean of HV in Centeral Island's Populations

1.0

125000000

Hypervolume

124900000

0.6 0.4

124800000

124700000 124600000

0.2

124500000 0.0 0 10

P) (G

IP)

V

0( 10

20

0(

G

P) 20

VI 0(

P)

) GP 0(

30

0 30

) IP (V

0 40

P) (G

0( 40

VI

P) 50

) GP 0(

0( 50

VI

P) 0 60

P) (G

0( 60

V

) ) ) ) ) ) ) ) ) P) IP) IP IP IP) IP GP GP GP GP GP GP VI 0 ( 0 0 (V 00 ( 00 (V 00 ( 00 ( V 00 ( 00 (V 0 ( 0 0 (V 00 ( 00 ( 30 6 2 5 4 6 2 1 5 3 4

IP) 10

Generation

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ5

95% CI for the Mean of HV in all Populations 124999500

0.8

124999000

Hypervolume

Coverage Rate

95% CI for the Mean of CR in the all Populations 1.0

0.6 0.4 0.2 0.0 0( 10

124998500

124998000

124997500

124997000 ) GP

V 0( 10

IP)

0( 20

GP

) 20

V 0(

IP

)

) ) ) ) P) P) P) P) GP GP GP GP VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0 0 0 0 0 0 4 5 30 6 5 60 3 4

Generation

0( 10

GP

) 0 10

(V

) IP

0 20

P (G

) ) ) ) ) ) P) IP) IP) IP IP GP GP GP GP VI 0( 00 (V 00 ( 00 (V 00 ( 0 0 ( V 00 ( 00 (V 0( 0 0 3 4 5 6 3 2 5 4 6

)

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ5 Fig. 33. Experiment result of running GP (multiple NSGA-IIs) and VIP (VIP-MOEA) on DTLZ5.

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

0.8

95% CI for the Mean of HV in Centeral Island's Populations

0.8

124700000

Hypervolume

124800000

0.6

0.4

124600000 124500000 124400000

0.2

124300000

0.0

124200000

) ) ) ) ) ) P) P) P) P) P) P) GP GP GP GP GP GP VI VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 10 20 40 30 50 60 10 40 30 20 50 60

0(

10

Generation

GP

) 0 10

IP

(V

) 20

0(

GP

) 0( 20

) ) ) ) ) ) P) IP) IP) IP IP GP GP GP GP VI 0( 00 (V 00 ( 00 (V 00 ( 0 0 ( V 00 ( 00 (V 30 4 6 5 5 3 6 4

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ7

95% CI for the Mean of CR in the all Populations

95% CI for the Mean of HV in all Populations

1.0

124350000 124340000 124330000

Hypervolume

Coverage Rate

0.8

0.6

0.4

0 10

124310000 124300000 124290000 124280000

0.2

0.0

124320000

124270000

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

95% CI for the Mean of CR in Center Island's Populations 1.0

124260000 P) (G

0( 10

P) VI

20

0(

) GP

0 20

(

P VI

)

) 0

30

P (G

0( 30

P) VI

0( 40

) GP

Generation

0 40

IP (V

) 0(

50

) GP

0( 50

P) VI

0( 60

) GP

0( 60

P) VI

0( 10

) ) ) ) ) ) ) ) ) ) ) IP) IP IP IP IP GP GP GP (VIP GP GP GP 0 ( 00 (V 00 ( 0 0 (V 00 ( 00 (V 00 ( 00 ( V 00 ( 00 (V 0 0 0 3 5 2 4 6 1 2 6 3 4 5

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ7 Fig. 34. Experiment result of running GP (multiple NSGA-IIs) and VIP (VIP-MOEA) on DTLZ7.

2885

2886

Refrence Point=[ 500 500 500]

95% CI for the Mean of HV in Centeral Island's Populations

95% CI for the Mean of CR in Center Island's Populations 1.0

135000000

Hypervolume

130000000

0.6

0.4

125000000

120000000 0.2 115000000

0.0 0( 10

GP

)

IP

)

V 0( 10

) ) ) ) ) P) P) P) P) P) GP GP GP GP GP VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0 0 0 0 0 0 0 0 0 4 6 20 3 5 3 2 4 5 6

0( 10

GP

) 0 10

I (V

P)

0(

GP

20

) 0( 20

VI

P) 30

0(

) ) P) P) P) P) P) IP) GP GP (VI (G VI VI 0 (G 0 ( 00 (V 00 0 0( 0( 0 0 0 0 0 6 4 5 3 4 6 5

Generation

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ1 Reference Point=[500 500 500]

95% CI for the Mean of CR in the all Populations

95% CI for the Mean of HV in all Populations

1.0

125000000 124750000

Hypervolume

Coverage Rate

0.8

0.6

0.4

124500000 124250000 124000000 123750000

0.2

123500000 0.0 0( 10

G

P)

0( 10

VI

P) 20

G 0(

P) 0 20

(V

I

P) 0

30

P) (G

0( 30

V

IP)

0( 40

G

P)

VI

0( 40

Generation

P)

0( 50

G

P)

0( 50

VI

P) 0 60

P) (G

0( 60

VI

P)

0( 10

GP

) 0 10

(V

IP

) 0( 20

GP

) 0( 20

VI

P) 30

0(

) ) P) P) P) P) P) IP) GP GP (VI (G VI VI 0 (G 0 ( 00 (V 00 0( 0 0( 0 0 0 0 0 5 6 4 5 4 3 6

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ1 Fig. 35. Experiment result of running GP (multiple MOEA/Ds) and VIP (VIP-MOEA) on DTLZ1.

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

0.8

95% CI for the Mean of HV in Centeral Island's Populations

0.8

124995000

Hypervolume

125000000

0.6

0.4 0.2

124985000

124980000

124975000

0.0 0 10

124990000

)

P (G

0( 10

P VI

) 20

GP 0(

) 20

0(

P VI

) 30

) GP 0(

0( 30

P) VI

0 40

P (G

) 0 40

IP (V

) 0

50

P (G

) 0( 50

P VI

) 0 60

P) (G

0( 60

P VI

) 0( 10

GP

) 0 10

(V

IP

) 0( 20

GP

) ) ) ) ) ) P) IP) IP) IP IP GP GP GP GP VI 0( 00 (V 00 ( 00 (V 00 ( 0 0 ( V 00 ( 00 (V 0( 6 4 5 30 6 4 3 5 20

)

Generation

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ2 95% CI for the Mean of HV in all Populations

95% CI for the Mean of CR in the all Populations 125000250

1.0

125000000

Hypervolume

Coverage Rate

0.8

0.6 0.4

124999750 124999500 124999250 124999000

0.2

124998750

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

95% CI for the Mean of CR in Center Island's Populations 1.0

124998500

0.0

) ) ) ) ) ) P) P) P) P) P) P) GP GP GP GP GP GP VI VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 30 40 50 20 10 60 30 20 60 10 50 40

Generation

) ) ) ) ) ) ) ) ) P) IP ) IP IP ) IP IP GP GP GP GP GP GP VI 0 ( 0 0 ( V 00 ( 00 ( 0 ( 0 0 (V 00 ( 00 (V 0 0 ( 00 (V 00 ( 00 (V 6 30 5 2 4 5 6 2 4 1 3

10

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ2 Fig. 36. Experiment result of running GP (multiple MOEA/Ds) and VIP (VIP-MOEA) on DTLZ2.

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95% CI for the Mean of HV in Centeral Island's Populations

95% CI for the Mean of CR in Center Island's Populations 1.0

130000000

Hypervolume

125000000

0.6

0.4

120000000

115000000 0.2 110000000

0.0

) GP

0( 10

0( 10

V

) IP 20

G 0(

P) 20

0(

V

) IP 30

0

P) (G

0( 30

V

IP)

0( 40

) GP

VI

0( 40

P)

0(

50

) GP

0( 50

V

) IP 0 60

P) (G

0( 60

V

) IP

0( 10

GP

) 0 10

(V

IP

) 0( 20

GP

) ) ) ) ) ) P) IP) IP) IP IP GP GP GP GP VI 0( 00 (V 00 ( 00 (V 00 ( 0 0 ( V 00 ( 00 (V 0( 6 4 5 30 6 4 3 5 20

)

Generation

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ3 Reference Point=[500 500 500]

95% CI for the Mean of HV in all Populations

95% CI for the Mean of CR in the all Populations 132500000

1.0

130000000

Hypervolume

Coverage Rate

0.8

0.6 0.4

127500000 125000000 122500000 120000000

0.2

117500000 115000000

0.0 0 10

P (G

)

) ) ) ) ) P) P) P) P) P) P) GP GP GP GP GP VI VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 40 30 50 20 60 10 40 30 50 20 60

Generation

) ) ) ) ) ) ) ) ) P) IP ) IP IP IP ) IP GP GP GP GP GP GP VI 0 ( 0 0 (V 00 ( 00 ( 0 ( 0 0 (V 00 ( 00 (V 0 0 ( 00 (V 00 ( 00 (V 6 30 2 5 4 6 2 5 4 1 3

10

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ3 Fig. 37. Experiment result of running GP (multiple MOEA/Ds) and VIP (VIP-MOEA) on DTLZ3.

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

0.8

95% CI for the Mean of CR in Center Island's Populations

Refrence Point=[ 500 500 500]

95% CI for the Mean of HV in Centeral Island's Populations

1.0

125005000 125000000

Hypervolume

0.6 0.4

124995000 124990000 124985000 124980000

0.2

124975000 124970000

0.0

) ) ) ) ) ) P) P) P) P) P) P) GP GP GP GP GP GP VI VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 10 50 40 60 30 20 60 40 20 50 10 30

0( 10

Generation

GP

) 0 10

(V

IP

) 20

0(

GP

) ) ) ) ) ) P) IP) IP) IP IP GP GP GP GP VI 0( 00 (V 00 ( 00 (V 00 ( 0 0 ( V 00 ( 00 (V 6 5 4 30 4 6 3 5

) 0( 20

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ4 95% CI for the Mean of HV in all Populations

95% CI for the Mean of CR in the all Populations

125000100

1.0

125000000

0.8

Coverage Rate

Hypervolume

Reference Point=[500 500 500]

124999900

0.6 0.4

124999800 0.2 124999700 ) ) ) ) ) ) P) P) P) P) IP) IP) IP IP GP GP GP GP (G VI VI 0 (G 0 ( 0 0 (V 00 ( 00 (V 00( 00 (V 00 ( 00 (V 00 0( 0( 60 5 3 2 4 10 60 50 2 4 1 3

Generation

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

0.8

0.0

) ) ) ) ) ) P) P) P) P) P) P) GP GP GP GP GP GP VI VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 40 50 60 10 30 20 10 30 50 20 60 40

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ4 Fig. 38. Experiment result of running GP (multiple MOEA/Ds) and VIP (VIP-MOEA) on DTLZ4.

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95% CI for the Mean of HV in Centeral Island's Populations

0.8

124980000

Hypervolume

124990000

0.6 0.4

124970000 124960000 124950000

0.2

124940000 124930000

0.0

) ) ) ) ) ) P) P) P) P) P) P) GP GP GP GP GP GP VI VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 10 30 50 20 60 40 10 60 20 30 50 40

0( 10

Generation

GP

) 0 10

(V

IP

) 20

0(

GP

) 0( 20

VI

P) 30

0(

) ) P) P) P) P) P) IP) GP GP (VI (G VI VI 0 (G 0 ( 00 (V 00 0 0( 0( 60 5 40 30 50 4 60

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ5

95% CI for the Mean of HV in all Populations

95% CI for the Mean of CR in the all Populations 125000000

1.0

Hypervolume

Coverage Rate

124999000 0.8

0.6 0.4

124998000 124997000 124996000 124995000

0.2

124994000 124993000

0.0 0( 10

G

P)

0( 10

V

IP)

G 0(

20

P)

0( 20

VI

P)

0(

30

) GP

0( 30

V

IP) 0 40

P) (G

0(

40

Generation

VI

P) 50

0

P) (G

0 50

(V

I

P) 0 60

P) (G

60

0(

V

) IP

) ) ) ) ) ) ) ) ) P) IP ) IP IP ) IP IP GP GP GP GP GP GP VI 0 ( 0 0 ( V 00 ( 00 ( 0 ( 0 0 (V 00 ( 00 (V 0 0 ( 00 (V 00 ( 00 (V 5 4 10 30 2 6 4 6 3 5 2 1

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ5 Fig. 39. Experiment result of running GP (multiple MOEA/Ds) and VIP (VIP-MOEA) on DTLZ5.

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

95% CI for the Mean of CR in Center Island's Populations 1.0

95% CI for the Mean of HV in Cente ral Island 's Popu lation s

95% CI for the Mean of CR in Center Island 's Populat ion s 1.0

124300000 124250000

Hypervolume

0.6 0.4

124200000 124150000 124100000 124050000

0.2

124000000 0.0 0 10

P) (G

0 10

) IP (V

) ) ) ) ) P) P) P) P) P) GP GP GP GP GP VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 30 20 60 50 40 50 60 20 30 40

0( 10

) GP

0( 10

) ) ) ) ) ) ) ) ) ) P) IP IP IP IP GP GP GP GP GP (VIP VI 0 ( 00 (V 00 ( 00 (V 00 ( 0 ( 00 ( V 00 ( 00 (V 0 20 50 3 4 6 40 3 2 6 5

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in Central Island on DTLZ7 95% CI for the Mean of CR in the all Popu lat ions

95% CI for the Mean of HV in all Populat ion s 124350000

0.8

Hypervolume

Coverage Rate

1.0

0.6 0.4

124300000

124250000

124200000

0.2

H. Rajabalipour Cheshmehgaz et al. / Applied Soft Computing 13 (2013) 2863–2895

Coverage Rate

0.8

124150000

0.0

) ) ) ) ) ) P) P) P) P) P) P) GP GP GP GP GP GP VI VI VI VI VI VI 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 60 30 50 40 20 10 50 60 40 20 10 30

0( 10

) GP

0 10

) ) ) ) ) ) ) ) ) ) ) IP IP IP IP IP GP GP (VIP GP GP GP (V 0 ( 00 ( V 00 ( 0 0 (V 0 ( 00 (V 00 ( 0 0 (V 00 ( 0 6 50 3 4 20 6 5 40 3 2

Generation

CR (left) and HV (right) in different generation gained by GP and VIP in all islands on DTLZ7 Fig. 40. Experiment result of running GP (multiple MOEA/Ds) and VIP (VIP-MOEA) on DTLZ7.

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Potential direct shipment Shipment Products flow

Supplier 1

DC1

Retailer 1

Customer 1

Supplier 2

DC2

Retailer 2

Customer 2

Supplier l

DCn

Retailer n

Customer k

Fig. 41. 4-level flexible logistic network with potential direct shipments.

Fig. 42. Known-PFs gained by multiple NSGA-II and VIP-MOEA on 4-fLN1 .

given by all islands is noticeable better than the corresponding value gained by GP, see Fig. 39.

7.2. Real-world combinational optimization problem: flexible logistic network design Optimal logistic networks for the supply of customers in due time at lowest costs, have been recognized such a competitive part for any productive or service business. The logistic network design is often considered as a long-term multi-objective optimization problem [22]. The case study here considers 4-level flexible logistic networks (4-fLN) comprised of potential suppliers (S), distributing centers (DC), retailers (R) and also actual customers (C) at the first level. Each customer has pre-specified demand of single item product for a period of time (e.g. season, year, etc.). The main objective is to calculate the status (decision on being open or close) of suppliers, distributing centers or retailers; and transportation links in order to minimize the lead time – the delay from the time of making an order until it is received – (time), transportation cost (TC) and facility costs (FC), simultaneously. With consideration that the network can be flexible with potential (probably expensive) direct shipments from: suppliers to DCs, suppliers to retailers, suppliers to customers, DCs to retailers and DCs to customers (see Fig. 41). Some more other options like capacitated constraints that are not considered here have been considered in other works (e.g. [23]).

Table 3 Information of two 4-fLN design problem. Case

# of suppliers

# of distributing centers

# of retailers

# of customers

4-fLN1

5

8

10

20

VIP-MOEAs benefiting from NSGA-II (NSGA-II&VIPs with their corresponding versions (multiple NSGA-IIs) are use and compared on a 4-fLN introduced in Table 3. The values of known-PFs (time, facility cost and transportation cost) gained by multiple NSGA-II and VIP-MOEA on 4-fLN1 by 300 iterations, are visually shown in Fig. 42. The number of these solutions in Central Island’s populations is 19 and 50 given by multiple NSGA-IIs and VIP-MOEA respectively. Moreover, multiple NSGA-II and VIP-MOEA run 20 times and the values of CR and HV, estimated with 95% CI on the cases. The results are shown in Fig. 43. 8. Scalability of the proposed island model To study how the number of VIP regions (or islands) affects the proposed algorithms, we investigate the scalability of the island model with different numbers of VIP regions (or islands). This number can be also a considerable factor to any parallel and distributed version of VIP-MOEA that is not discussed here. Therefore, we have to test four and 25 VIP regions (or islands) as the two nearly extreme

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2893

Refrence Point=[ 100000 100000 100000]

95% CI for the Mean of HV in Centeral Island's Populations

1.0

8.0000E+14

0.8

7.0000E+14

Hypervolume

Coverage Rate

95% CI for the Mean of CR in Center Island's Populations

0.6 0.4

50

5.0000E+14

4.0000E+14

0.2 0.0

6.0000E+14

3.0000E+14 P) (G

( 50

P) VI

0

10

P (G

) 10

0(

P VI

) 15

G 0(

P)

P) VI

0( 15

0 20

P) (G

0

20

IP (V

) 0 25

P (G

) 0( 25

P VI

)

) P) GP VI 0( 0( 30 30

( 50

GP

) ( 50

VI

P)

GP

0( 10

)

P)

VI

0(

10

0(

15

GP

)

VI

0( 15

P)

) ) ) ) P) P) IP IP GP GP (G VI 0 ( 00 (V 50 ( 50 (V 00 0( 2 3 20 30 2 2

Generation

Generation

CR (left) and HV (right) in different generation gained by Multiple NSGA-IIs and VIP-MOEA in Central Island on 4-fLN1 Reference Point=[ 100000 100000 100000]

95% CI for the Mean of HV in all Islands' Populations

1.0

8.0000E+14

0.8

7.0000E+14

Hypervolume

Coverage Rate

95% CI for the Mean of CR in all Islands' Populations

0.6 0.4

6.0000E+14 5.0000E+14 4.0000E+14

0.2

3.0000E+14

0.0

) ) ) ) ) P) P) P) P) P) P) IP) GP GP GP GP GP VI VI VI VI (G VI (V 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 50 50 30 25 20 10 15 30 20 25 10 15

Generation

50

P (G

)

) ) ) ) ) ) ) ) ) ) IP) IP IP IP IP IP GP GP GP GP GP (V 0 ( 00 (V 50 ( 50 (V 00 ( 00 (V 50 ( 5 0 (V 00 ( 00 (V 50 2 2 3 10 1 1 2 2 1 3

Generation

CR (left) and HV (right) in different generation gained Multiple NSGA-IIs and VIP-MOEA in all islands on 4fLN1 Fig. 43. Experiment result of running multiple NSGA-IIs and VIP-MOEA on 4-fLN1 .

numbers of individual VIP regions (regarding to the populations size in Central Island, 50). The algorithms run on DTLZ1 with the implementation of VIP-MOEA with NSGA-II in its Central Island. All the parameter settings are the same as Section 7.1 and Experiment 2, except the number of VIP regions and islands. According to the results, there is no considerable change on CR in all circumstances; however, VIP-MOEA still gives the better CR in both cases as compared with multiple NSGA-IIs case. Only HV shows a difference in values as the number of VIP regions in VIPMOEA or islands in multiple NSGA-IIs increased. The values of HV in the both cases (GP: multiple NSGA-II, VIP: VIP-MOEA) and in different generations (iterations) are presented in Fig. 44. Clearly, HV reaches a significant value in Central Island by VIP with 25 VIP regions by 100 iterations comparing with the similar case with four VIP regions (see Fig. 44a and b – left). Although, VIP and GP in their Central Islands reach the same values of HV in their final generations (reaching just under 125,000,000), GP in the case with 25 islands, shows a sharp increase in reaching the value as compared with GP in the case with four islands. However, in overall, VIP shows better performance in Central Islands gaining higher values of CR during early generations, and nearly equal with GP as the generation goes more forward (see Fig. 44a and b – left).

The same analysis of HV gained from solutions in all islands, shows the obvious fact that the case with 25 islands or VIP regions reach to the value (just under 12,500,000), faster than the case with only four islands or VIP regions (see Fig. 44a and b – right). However, VIP is better than GP in the case with more islands based on the gained HV in all islands in different generations, particularly in early ones. The larger value in HV shows the more diversity among knownPFs, and vice versa. The lower values of HV gained by VIP with four VIP regions (or islands) in early generations should be due to the nature of environmental selection (VIP-ES) using by VIPMOEAs. The selection that defined in early sections, first tries to select knownPFs inside the respect VIP region. It might not concern about the diversity of these solutions in early generations, because the number of these solutions might not be enough, and then it is not necessary to do the proposed local ordering in the first group of solutions (see Fig. 4). In other words, the number of these solutions is less that the size of VIPMOEA’s population. As the generation continues, the number of these solutions might be increased and then the diversity of these solutions will be considered afterward. Whereas, with more divisions with VIP regions in the objective space, the regions are thinner, and the solution in VIP regions will be much closer to each other. Therefore, the diversity among the known-PFs

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Reference Point=[100000 100000 100000]

95% CI for the Mean of HV in all Populations

125000000

125000000

124999000

124999000

Hypervolume

Hypervolume

95% CI for the Mean of HV in Centeral Island's Populations

124998000

124998000

124997000

124997000

124996000

124996000

) ) ) ) ) ) ) ) P) IP IP) IP) IP IP IP) GP GP GP GP GP 0 ( 00 (V 00 ( 00 (V 00 ( 00 (V 00 (G 00 (V 00 ( 00 (V 00 ( 00 (V 5 4 6 3 10 2 5 3 1 2 6 4

0( 10

) GP

IP)

0 10

(V

0 20

) ) ) ) ) P) P) P) IP) IP) IP IP GP GP GP (G VI 0 ( 00 (V 00 (G 00 (V 00 ( 00 (V 00 ( 00 (V 0( 5 4 6 30 5 4 3 6 20

Generation

Generation

a)

HV gained in Central Island (left) and all islands (right) by multiple NSGA-IIs and VIP-MOEA with four islands and VIP regions on DTLZ1

Reference Point=[100000 100000 100000]

Reference Point=[100000 100000 100000]

95% CI for the Mean of HV in all Populations

125000000

125000000

124999000

124999000

Hypervolume

Hypervolume

95% CI for the Mean of HV in Centeral Island's Populations

124998000

124997000

124996000 0 10

124998000

124997000

) ) ) ) ) P) P) P) P) P) P) P) IP GP GP GP GP VI VI VI VI VI (G (G 0 ( 00 ( 0 ( 00 (V 00 ( 00 ( 0 ( 00 ( 0 ( 200 0( 40 30 50 6 10 5 20 6 3 4

Generation

124996000

) ) ) ) ) ) ) P) P) P) P) P) IP GP GP GP GP GP GP VI VI VI VI VI 0 ( 00 ( 0 ( 00 (V 00 ( 00 ( 0 ( 00 ( 0 ( 00 ( 0 ( 00 ( 6 30 50 40 20 10 3 4 1 2 6 5

Generation

b) HV gained in Central Island (left) and all islands (right) by multiple NSGA-IIs and VIP-MOEA with 25 islands and VIP regions on DTLZ1 Fig. 44. HV values gained in Central Islands and the whole island models with different number of VIP regions (islands).

collected from all islands, and in Central Island is unintentionally upheld. 9. Conclusion Division techniques were used to utilize evolutionary searches by multiple MOEAs, which locally explore their respect division effectively. These MOEAs might generate and then ignore some solutions, which are located outside their respect divisions. Hence, some new solutions with good quality might do not contribute into the next generations, because of the hard constraints created by the divisions. Moreover, the most efficient algorithms that benefit from multiple local-MOEAs rely on such frequent acts with noticeable overhead, particularly in parallel systems, to preclude any worthless exploration on infeasible regions. First, this research has proposed an effective local MOEA, VIPMOEA that explores not only inside the assigned division (a region in objective space) but also nearby the division. In fact, the division establishes such a soft constraint in objective space for VIPMOEA. Even with a division having no (true) PFs inside, the algorithm is able to search the nearby for PFs. Because it uses a particular mate selection to benefit more from inside solutions in the next generations, while the outside non-dominated solutions can be saved

in the next population as well through a sophisticated environmental selection. We tested VIPMOEA on DTLZ1 and DTLZ5 with different objective space divisions (called regions) that created by a polar-coordinate-based technique. Our illustrative tests have demonstrated how the algorithm works in two different situations: having PFs and having no PFs inside the regions. Second, we proposed an island model in order to benefit from multiple VIPMOEAs exploring independent regions in objective space. The model is made of one island, Central Island, for a conventional MOEA and multiple VIP Islands running their own VIPMOEAs and directly linked to Central Islands. The model is used to improve conventional MOEAs like NSGA-II and MOEA/D through a novel immigration strategy and an adjusting mechanism to change the search direction, slightly, in VIPMOEAs. Meanwhile, a polar-based division technique was designed to split the objective space among VIPMOEAs, effectively. Our analysis has shown that the proposed island model gained better coverage rate and hypervolume in the most DTLZ test cases as compared with the clustering MOEA and similar island models. With an application, the proposed model has gained definitely better quality known-PFs on a real-world combinational optimization problem; flexible logistic network design. We have also investigated the scalability of the model through a simple experiment with two different numbers of VIP regions (or islands). We have found that the proposed island model can reach

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better diversity of known-PFs and higher value in hypervolume, earlier, as the number of VIP regions in objective space increases. However, more VIP regions have no significant influence on another performance metric, coverage rate. Division and adjusting the divisions of objective space along multiple MOEAs has been proven to be very successful in MOPs. Our work in this research provides polar-based divisions of objective space given to the multiple novel local-MOEAs to search inside and nearby the assigned regions (divisions), effectively and efficiently. New development in division and other techniques in adjusting the regions (VIP regions) can be readily integrated with MOEA in the framework of the proposed island model and VIPMOEAs. Acknowledgments The authors are grateful to Farahnaz Kazemipour, the anonymous associate editor, and anonymous referees for their insightful comments. The authors would like to thank the Universiti Teknologi Malaysia (UTM) and Ministry of Higher Education (MOHE) Malaysia for giving financial support under the Research University Grant (Vot No. Q.J. 130000.7128.00J96), and also to the Research Management Centre (RMC) – UTM for continuous assistant and support of this research project. References [1] C.A.C. Coello, Evolutionary multi-objective optimization: a historical view of the field, IEEE Computational Intelligence Magazine 1 (2006) 28–36. [2] K. Deb, Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons Ltd., West Sussex, England, 2001. [3] E.G. Talbi, S. Mostaghim, T. Okabe, H. Ishibuchi, G. Rudolph, C.A.C. Coello, Parallel approaches for multiobjective optimization, Multiobjective Optimization: Interactive and Evolutionary Approaches 5252 (2008) 349–372, 470. [4] C.A.C. Coello, G.B. Lamont, D.A.V. Veldhuizen, Evolutionary Algorithms for Solving Multi-objective Problems, second ed., Springer Science + Business Media, LLC, New York, NY, 2007. ˜ C.A.C. Coello, E. Mezura-Montes, pMODE-LD + SS: an effective [5] A.A. Montano, and efficient parallel differential evolution algorithm for multi-objective optimization, in: PPSN’10 Proceedings of the 11th International Conference on Parallel Problem Solving from Nature. Part II, Springer-Verlag, Krak, Poland, 2010. [6] J. Branke, H. Schmeck, K. Deb, M. Reddy, Parallelizing multi-objective evolutionary algorithms: cone separation, in: CEC2004: Proceedings of the 2004 Congress on Evolutionary Computation, vols. 1 and 2, 2004, pp. 1952–1957, 2371.

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