Interactive Concept-Based Search using MOEA: The ... - CiteSeerX

International Journal of Computational Intelligence Volume 2 Number 3

Interactive Concept-Based Search using MOEA: The Hierarchical Preferences Case Gideon Avigad, Amiram Moshaiov, and Neima Brauner Section III, deals with hierarchical representation of concepts and their hierarchy-based objective-subjective evaluation. In addition, section III, together with the appendix, provides details on the evolutionary computation approach, which is used here to solve the presented interactive concept-based problem. Section IV provides examples to highlight the ideas of this paper and to demonstrate the suggested algorithm. Furthermore, it includes a discussion on some limitations of the current algorithm and suggests future work to ensure the appropriate development of a basic algorithm for conceptbased multi-objective problems. Finally, section V summaries the paper.

Abstract—Our recently introduced technique for multi-objective search of conceptual solutions, based on interactive evolutionary computation, is extended. The extension deals with concepts, which are represented by hierarchical trees of sub-concepts. The survivability of solutions is influenced by both model-based fitness and subjective human preferences. The concepts’ preferences are articulated via the hierarchy of sub-concepts. The suggested method produces an objective-subjective front. An academic example is employed to demonstrate the proposed approach.

Keywords—Conceptual solution, multi-objective search, sPareto, objective-subjective front, conceptual design, hierarchical human-preferences, hierarchical planning.

II. BACKGROUND This section describes the difference between the traditional multi-objective optimization problem, and the concept-based one. It further provides a background on the interactive concept-based search problem, and its solution by MOEA.

I. INTRODUCTION ECENTLY, a concept-based Interactive Evolutionary Computation (IEC) approach has been introduced for the exploration of conceptual solutions in multi-objective engineering design problems [1]. The use of conceptual solutions improves human-machine interface, and enables evaluating concepts, rather than just specific solutions, while taking into account human perceptions and subjective preferences. Dealing with engineering design problems, a progressive goal approach has been taken, where solution concepts are evolved around a dynamic target in a multiobjective space [1]. In [2] interactive evolution of conceptual solutions for a multi-objective path-planning problem has been investigated using a Pareto-directed, rather than a progressive goal approach. The Pareto-directed approach allows a nonlocalized inspection of solutions with respect to the objective space. Here, the Pareto-directed technique is described in more details, extended, and generalised to deal with problems involving conceptual solutions that are represented by a hierarchical tree of sub-concepts (sub-solutions). Such a hierarchical representation is useful in problem reduction and in application areas such as engineering design (e.g., subsystems), and hierarchical planning. The rest of this paper is organized as follows. The next section (II) provides the necessary background for understanding the subject of interactive concept-based search using a Multi-Objective Evolutionary Algorithm (MOEA).

R

A. Traditional Multi-Objective Optimization In the common Multi-objective Optimization Problem (MOP), such as described in [3-5], the set of Pareto optimal solutions is sought from the set of all possible particular solutions. Any particular solution is characterized by specific values of the problem variables representing a point in the problem variable space. The set of Pareto optimal solutions is found by comparing the performances of all particular solutions in the objective space for non-dominancy. The representation, in the objective space, of the set of nondominated solutions is known as the Pareto front. Finding the performances of particular solutions is usually done by the use of models. Evolutionary computation methods appear to be suitable for obtaining the Pareto-set due to their parallel search nature. This seems to be manifested in the current development and the popularization of Pareto-based MOEA solution techniques, which are surveyed and described in references such as [3-5]. B. Concept-Based Multi-Objective Optimization In contrast to the traditional MOP, in a concept-based MOP, such as dealt with here and in similar investigations (e.g. in [1, 2, 6, 7]), the interest is not on particular solutions and their performances, but rather on the performances of conceptual solutions. Commonly the notion of a conceptual solution is associated with abstractive ideas generated by humans. It describes a generic solution to a problem. During a conceptual solution stage, such as initial planning and conceptual design, no particular solution is stated. By a particular solution we mean a fully detailed solution, such that it has a one-to-one

Gideon Avigad is with the Iby and Aladar Fleischman Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel (e-mail: [email protected]). Amiram Moshaiov is with the Iby and Aladar Fleischman Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel (e-mail: [email protected]). Neima Brauner is with the Iby and Aladar Fleischman Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel (e-mail: [email protected]).

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particular those that are involved in the search for solutions when there is uncertainty and poor definition of the problem at hand. Such a situation is common in conceptual design. The first IEC treatment of the concept-based MOP has been suggested in [1,2]. The interest is in supporting humans, such as planners or designers, by computers, during concept search and selection. The search concerns conceptual solutions that can be represented by sub-sets of the set of particular solutions of the problem (see examples in [1,2]). It is further assumed that the performances of each particular solution are computable via models (e.g., tables, parametric models). Each conceptual solution, and its associated particular solutions, may be characterized by different models and/or range of variables, and consequently may possess different performances. The computation of merits by models might only partially reflect all issues that are involved in selecting a concept. Difficulties in realizing solutions associated with a particular concept, might not be modeled (e.g., difficulties of manufacturing in design problems, un-modeled hazards associated with conceptual plans). The interactive conceptbased search approach allows modifying the concept-based MOP such that, in addition to evaluating concepts by computable models, human preferences toward concepts and/or sub-concepts (S-Cs) can be included in an interactive fashion (e.g., [1, 2]). The interactivity element is significant for dealing with complex real-life situations. In many such problems the available models are limited (In fact, in some problems, such as in artistic design, the search is commonly dependent solely on human subjective preferences, as described in [8]). The interactive concept-based search, as done here and in [2], using MOEA, is based on modifying the common Paretobased approach, into a Pareto-directed approach. According to the later approach human preferences are combined with a Model-Based Fitness (MBF), such that evolution is affected by both the objective performances and the subjective preferences. Thus a fitness of a solution is influenced by its performances and by the preferences towards S-Cs, as related to the concept to which the solution belongs, and/or by concept preferences. The fitness, which results from considering both influences, is termed Human Machine Fitness (HMF). The presentation of solutions in the objective space, as obtained by the HMF using a ranking procedure, has been termed the objective-subjective front [2]. It should be noted that this ‘front,’ might not necessarily possess the nondominancy characteristics in the objective space. This is due to the fact that it is not based on performances alone, but also on the human preferences. Fig. 1 depicts the individual fronts of three concepts (red, blue and green curves for concept # 1, 2, and 3, respectively). Each curve represents the set of points in the objective space, which is associated with the Pareto set that is a sub-set of the associated particular solutions of the concept. These fronts can be individually obtained, as done in [6], by using an objective evaluation of the individuals solutions associated with the concept. The concept-based (combined) front (bold green-blue curve) is the solution to the objective evaluation (MBF alone) of the concept-based MOP of the three concepts. When HMF

relationship with a point in the objective space. In the conceptbased MOP multiple particular solutions are associated with a conceptual solution, constituting a one-to-many relationship between a conceptual solution and the objective space. The concept-based MOP has been recently formalized, including an associated notion of s-Pareto front, by Mattson and Messac (see [7]). C. Concept-Based Search using MOEA The use of MOEA in conjunction with concepts, which are represented as sets of particular solutions, as formalized by the concept-based MOP in [7], is relatively recent. Andersson, [6], employed MOEA to separately evolve concepts by way of their particular solutions. The novelty of our MOEA approach, as done in [1-2], and extended here, is in the ability to simultaneously evolve several concepts. Moreover, the approach provides a method to combine human preferences and takes into account the hierarchy of sub-conceptual solutions. This is further discussed below. In solving a concept-based MOP it is commonly assumed that models to evaluate the associated particular solutions exist. In such a case a Pareto front can be found for each concept, independently, by its associated sub-set of particular solutions. Eventually, all such Pareto fronts can be further organized to produce one final front (e.g., [6]). Our approach differs from such a sequential approach by evolving the concepts simultaneously. The simultaneous approach allows interactive exploration of concepts (as detailed in the next subsection). It should be noted that the simultaneous approach is computationally more efficient. This stems from the fact that no full development of fronts of bad concepts is expected. In contrast, the sequential technique inherently involves such a development. Moreover, the simultaneous approach is an essential part of the Pareto-directed approach to obtain the objective-subjective front, as explained in the review below. D. Interactive Concept-Based Search using MOEA In ‘real-life’ situations humans rely on their experiences and preferences in choosing a conceptual solution, and eventually they translate the chosen concept into a chosen particular solution. This is usually done with or without the ability to explicitly evaluate the merits of the chosen particular solution. The articulation of human preferences in MOEA is not new. In [4], Van Veldhuizen and Lamont review this topic and categorize the articulation of preferences into three categories characterized by the stage at which the articulation of preferences is performed including the a-priori, progressive and posteriory methods. The progressive methods involve interactivity between humans and computers during the decision making process. Takagi, [8], reviewed the use of Interactive Evolutionary Computation (IEC). IEC approaches might be categorized by the level of explicitness of the human intervention in the evolutionary process [9]. Located on the implicit end of the spectrum are the automatic, model-based algorithms, and on the other (explicit) end are algorithms where the evolution is solely driven by human evaluations. The explicit category corresponds to what has been referred to as the narrow sense of IEC ([8]). A recent paper by Parmee [10] reviews different approaches for interactivity and in

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It is important to note that the current representation of the conceptual solution space is assumed to be a-priori to the search. This means that in the current implementation, the tree is not generated, while searching, as in common AI problems, and the search is limited for innovative rather than creative concepts.

Objective 2

Front of Concept #3 Front of Concept #1

Front of Concept #2

B. Hierarchy-Based Evaluation of Concepts In the concept-based MOP the explicit interest is in the performances of conceptual solutions. For this purpose a representation of the concepts’ performances, in the objective space, is sought (as explained in section II). A parallel search process is performed, where a finite set, S, of particular solutions, is examined simultaneously for their relative performances. To carry out the search, an evaluation of each particular solution has to be performed. The evaluation procedure is detailed below.

Objective 1 Fig. 1 Concept fronts and concept-based front

is employed the objective-subjective front can be any combination of the parts of the above curves, depending on the subjective influence. For example, the red curve could be the objective-subjective front when the decision makers strongly prefer S-Cs related to concept # 3, and not those related to the other concepts. The reader is referred to the demonstration of objective-subjective fronts in section IV. Preferring a concept can be done either by direct weighting of each concept, or indirectly through its S-Cs. In [1, 2], we dealt with the non-hierarchical case of concept representation by its S-Cs. Here we deal with a modification to include a hierarchy-based weighting as detailed in the next section.

Machine based fitness (MBF): Evaluating a particular solution of S is done by a multi-stage process, which is described below. First we employ a sorting and ranking procedure, following [8], to obtain ranked sets of nondominated solutions out of S. This is done purely by a modelr

based evaluation. Next we assign ‘dummy fitness,’ fitU , to each of the particular solutions, according to their rank, r, and further correct it by two penalty functions (see appendix for details). Subtracting from the initially assigned dummy fitness,

III. METHODOLOGY In this section the foundations for the interactive conceptbased search problem, which involves a hierarchy of subconcepts and their preferences, are outlined.

SC3

SC4 SC5

MBF

(see

i

= fit Ur − φ ir , m − m i

r ,m

(1)

Front-based concept sharing is applied to preserve concept diversity, and to prevent from a good concept to hinder the evolution of other potential concepts within a front. In-concept front niching preserves the diversity of particular solutions within each concept belonging to a particular non-dominated front (rank). The following explains the influence of each penalty in equation 1 using Fig.s 3 & 4. In Fig. 3, two nondominancy ranks, within the objective space, are shown. Each rank contains several individuals. The individuals’ of each concept are designated by the same symbol (circles, crosses, and triangles for three depicted concepts). For the sake of explanation several of the depicted individuals are indexed by a number, and are referred to in Fig. 4.

SC2

SC6 SC7 SC8 SC9

φir ,m ,

equation A5 of the appendix), and further subtracting an inconcept front niching penalty, mi r,m , (see equation A9 of the appendix) results in an MBF of the i-th individual of the r-th rank:

A. Hierarchical Representation of Concepts A hierarchical ‘AND/OR’ tree is used to represent the set of all conceptual solutions of the problem (conceptual solution space). Each concept is represented as a hierarchical ‘AND’ tree, which is extracted from the ‘AND/OR’ tree, by decisions taken at the ‘OR’ nodes. An ‘OR’ node is termed D-N (decision node), indicating that a decision on selecting a branch at that node has to be taken for the extraction of a concept. The ‘AND’ nodes of the ‘AND/OR’ tree can be viewed as sub-conceptual solutions (S-Cs) of the problem. Fig. 2 depicts such a tree. It includes 10 S-Cs, which are used to express 8 different concepts, as further detailed in section IV.

SC1

fitUr , a front-based concept-sharing penalty,

SC10

Fig. 2 Conceptual solution space representation

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S-C). In such a case a high preference towards a bolt arrangement to mount the ‘delta’ wing is irrelevant. The decision makers may assign weights to some S-Cs of the problem, with values in the interval [-1, 1], where -1 designates pure dislike, and 1 stands for highest preference. For each ‘AND’ tree, the nodes, which represent weighted SCs, are assigned with the respective weights. Branches, below any of the nodes, which represent S-Cs with assigned preferences, are pruned. The S-Cs of the pruned tree, which have no preference, are automatically assigned with zero weights. The weights, of the resulting pruned tree, are used to obtain the m-th concept-weight, Hm, representing the concept preference. Starting from the leaves of the pruned tree, the weight, w(pr), of each parent node, is calculated by averaging the weights of its children, w(ch). The weight of a parent node is:

Fig. 3 Objective space presentation

w (pr ) =

1 nL

nL

∑ w(ch) n =1

(2) and H m = w(root ) where, nL is the number of the node’s children. The calculation of the weight of the m-th concept, H m = w(root ) , is obtained by calculating the weights of the ancestors up to the root node of the ‘AND’ tree of the m-th concept. Finally, each i-th individual of the m-th concept is assigned with Hi = Hm. The following example illustrates the procedure. Fig. 5a depicts an extracted ‘AND’ tree, with some of its nodes assigned with weights (the numbers by the nodes). Fig. 5b shows the pruned tree. The weight of the concept is obtained as follows: Hi = ((0.6 + 0.0 + 0.0)/3 +0.3)/2 = 0.25.

Fig. 4 MBF for selected individuals Individuals 1-5, of the first rank (rank 1), are assigned 1

initially with the fitness fit U (see Fig. 4). Individual #3 has the smallest reduction of fitness, in comparison with other members of the front. This is because it belongs to a concept with the smallest number of members in rank 1, and is not as niched as individual 5. Individual 2 belongs to the most populated concept in the first rank, and is more niched in comparison with other individuals of the rank. Therefore, it is the most penalized individual in the rank. Individuals of the

0.3

0.3

2

second rank, such as 6-9, are dummy fitted with fit U and further re-evaluated by their front-based concept sharing, and in-concept front niching, within the 2nd rank. This is demonstrated by Fig. 4, as resulting from the individual distribution in rank 2, which is depicted in Fig. 3.

-0.3

0.6

0.0

0.6 0.0

0.8 0.1 Fig. 5 (a) Concept’s ‘AND’ tree (b) The pruned tree

Hierarchy-based weighting: The process of evaluating a particular solution is not completed without the insertion of the affect of humans on the evaluation. The fitness of a particular solution should be corrected according to the human preference of its associated concept. The team’s preferences of S-Cs are accounted for in accordance with their location in the ‘AND’ hierarchical tree of each of the concepts. The following procedure ensures that human preferences of S-Cs are not contradicting to the hierarchy of the S-Cs. Preferences of S-Cs are not to be considered whenever preferences exist at ancestors’ nodes. For example, consider the case when the designers of an airplane reject using a ‘delta’ wing (lowest preference for this

Human-machine fitness: As described in the background section, the human preferences should be combined with the MBF so that evolution is affected by both the objective performances and the subjective preferences. Thus the fitness of a solution is influenced by its performances and by the preferences of the decision makers towards the S-Cs, as related to the concept to which the solution belongs. The fitness, which results from considering both influences, is termed Human Machine Fitness (HMF), where HMF = f (MBF, H). The results presented in section IV, were obtained using the following bi-linear HMF function for the i-th individual.

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HMFi =

fiti ⋅ ( H i + 1)

fit i + ( fit max − fit

TABLE I ALGORITHM'S PARAMETERS

(3)

for − 1 ≤ H i ≤ 0 max, m i

) ⋅ (H i )

parameter δ

for 0 < H i ≤ 1

ε

np

Δ

where, fitmax is the maximal machine fitness over all individuals within the generation, and fit imax, m is the maximal fitness over all individuals belonging to a concept m of the generation.

α P pc pm

C. MOEA Implementation The concept-based MOEA implementation, as done in [1,2], and in related studies [11, 12], involves Compound Individuals (C-Is) holding structured genetic codes. Decoding a C-I results in a particular solution, which is related to a concept. The pseudo MOEA for the interactive concept-based multi-objective search is outlined below.

value 40 10 250 40 1.2 8 0.7 0.03

A. Non Hierarchical Test Case The following problem demonstrates the topics of conceptbased front, and the objective subjective front by a nonhierarchical academic example. The objectives, used in this academic example, are: f1 = x 2 + 5b + 10

(4)

f 2 = ( x − 4) 2 + c (b − 1) x

Pseudo code for the interactive concept-based MOEA INITIALIZE: C-I= compound individuals While decision-making continues Interactively assign S-Cs’ preferences While generation ≤ final generation While not all individuals’ fitness computed Decode C-I Compute Performances Compute MBF (pseudo-code A.1 in the appendix) Compute H (equation. 2) Compute HMF (equation. 3) Re-compute fitness (Pseudo code A.2 in the appendix) End C-I = Reproduce C-I C-I = recombine C-I with pc probability C-I = Mutate C-I with pm probability End Introduce objective-subjective front End

The problem domain, x, is defined within the interval [-10, +10]. The space consists of four concepts, which are outlined in table 2. The concepts are composed of four S-Cs, which are characterized by: c=-3, c= -4, b=+ 2 and b=+3. TABLE II SUMMARY OF CONCEPTS AND THEIR LEGEND

Concept # 1 2 3 4

c values -3 -4 -4 -3

b values +2 +2 +3 +3

Symbol circle star rhombus plus

The results of the first simulation are shown in Fig. 6, involving no human preferences towards the S-Cs. The surviving particular solutions, which are related to the concepts (designated as different symbols), constitute the concept-based front. The curves in Fig.s 6 & 7, represent the non-surviving parts of the fronts of the individual concepts. It can be seen that concept # 1 (plus) did not survive at all, while the others share the resulting front. At the top part of the front, concept-based front sharing allows the mutual survival of concepts # 1 & 2 (circles & stars). Without such sharing the evolution commonly results in an upper part of the front consisting of solutions that are related to just one of the concepts (#1 or #2). Furthermore, in-front concept sharing enhances the spreading of the surviving solutions along the front. The middle part of the front of Fig. 6 contains only one dominating concept (stars), and at the lower part, it contains another concept (rhombus).

It is noted that due to the possible shuffling of the initially assigned ranks, by the human intervention, the HMF of a large portion of the population may rise rapidly. This can cause exploitation at a too early stage of the search. Therefore a reranking (a rescaling procedure) procedure that increase the search pressure is implemented (see appendix for details). The outlined MOEA is used in the following numerical study. IV. NUMERICAL STUDY The numerical study is divided into three sub-sections. Subsection A demonstrates the fundamentals of the interactive concept-based MOEA. In sub-section B, a test case is used to show the affect of the hierarchical preferences on the evolution and on the resultant objective-subjective front. Subsection C, discusses the limitations of the algorithm and suggests some improvements for future work. The parameters of the genetic algorithm, used here, are detailed in Table I, where pc, and pm, are the probabilities for crossover and mutation, respectively.

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preferences on the resulting front. The bi-objectives used here are: f1 = x 2 + b ⋅ c f 2 = ( x − 2) 2 + b ⋅ d

(5)

The parameter, x, is the problem parameter searched within the interval [-10, +10]. Eight concepts are evolved based on ten S-Cs. The S-Cs, within the hierarchy arrangement of the ‘AND/OR’ tree, are depicted in Fig. 2. Each of the 8 concepts can be represented by an ‘AND’ tree that is extracted from that ‘AND/OR’ tree. Table III provides a list of concepts and their associated symbols (legend) as used in the following figures.

Fig. 6 Concept-based front To demonstrate the affect of human preferences in the nonhierarchical test case, the following situation is examined. The weight of the S-C characterized by c= -3 is assigned with a value of 1.0 (the highest preference possible), and the weight for c= -4 is chosen as –0.6. These preferences cause the concepts, associated with c=-3 (concepts 1 & 4), to be preferred over those associated with c=-4 (concepts 2 & 3). In this case, which is a non-hierarchical case, the Hm of concepts # 1 and 4 is 0.5 (as a result of the average of 1 and 0.0) while these of concepts # 1 and 3 become -0.3 (as a result of the average of -0.6 and 0.0). This follows an averaging procedure that has been suggested in [1, 2].

TABLE III SUMMARY OF CONCEPTS AND THEIR LEGEND

The parameters b, c, and d, which are associated with the SCs, dictate different performance models, (as related to equation 5), for the various concepts. It should be noted that SC1 to S-C10 are related to b=1.0, b=1.5, c=2.0, c=3.0, d=2.0, d=1.0, c=1.0, c=2.0, d=0.5, and d=1.5, respectively. It is also noted that the equal values of c=2.0 for S-C3 and S-C8 mean that these S-Cs are actually the same, and the different indices is a result of the representation. The problem parameter 'b' characterizes the highest S-Cs of the hierarchy, and the rest of the parameters are associated with the S-Cs of the lower hierarchy. Deciding on the branch at each D-N, leads to an ‘AND’ tree of S-Cs, which corresponds to a concept. Each concept of this example has its unique model resulting from determining the values of the parameters b, c, and d. Fig. 8 shows a part of the initial population. The eight concepts are distributed in the objective space according to their performances, as calculated by equation 4. Fig. 9 depicts the resulting front. It shows that three concepts survived (concepts # 5,7,8). The surviving concepts are associated with the S-C2 of b=1.5. This front is the concept-based front that is achieved by an evolution that is influenced just by the objective, machine-based fitness. Fig. 10 shows the resulted front with preference assignment. A weight of 0.6 for the S-C1 (associated with b=1), is used. Clearly, this causes a change from the front of Fig. 9. In addition to concept # 5, which belongs to the branch of S-C2 (b=1.5), a second concept (concept # 1) survived belonging to the branch of S-C1 (with b=1). It is noted that any assigned weights for S-C3 ÷ S-C6 (while S-C1 is also assigned with a weight), will not be accounted due to the pruning procedure.

Fig. 7 Objective-subjective front

The results of the evolution are shown in Fig. 7 (the curves are left as a reference). Fig. 7 is a result of a Pareto-directed interactive concept-based evolution, which is achieved by preferring different S-Cs. Comparing Fig. 7 with Fig. 6, it can be seen that the winning concepts are different. This is due to the influence of the subjective preference articulation. Two out of the three surviving concepts in the concept-based front disappeared (concepts #2 and #3). A new concept has survived (concept #4) and it is a part of the objective-subjective front. It is noted that under these specific preferences the objectivesubjective front is the one that the decision makers are introduced with. B. Hierarchical Test Case The purpose of the following bi-objective academic example is to demonstrate the ideas presented in this paper and in particular to show the affect of the hierarchical

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Fig. 11 S-C6 weight= 0.6

Fig. 8 Initial distribution

Fig. 11 shows the results, with a change of preference of a S-C, which belongs to a lower hierarchy. S-C6 (associated with d=1) is assigned a preference weight of 0.6. This assignment causes the survival of concept # 1, yet the resulting front is not as full, in comparison with that of Fig. 10. This is due to the lower location of the preferred S–C within the hierarchy. When the S-Cs, associated with c=2 and d=1, are both assigned with weights of 0.6, the result is similar to the one depicted in Fig. 10. This is because the weight of concept #1 is the same as it was in the case that led to the results of Fig. 10. Further increasing the preference towards S-C1 (the b=1 branch) to 1.0, a further increase in the survivability of concepts, which include S-C1, takes place, as shown in Fig. 12. Fig. 9 Machine-based front

Fig. 12 S-C1 weight=1.0

Three out of the four concepts, belonging to that branch (b=1), appear in the resulting objective-subjective front of Fig. 12. The forth concept did not appear due to its low performance. If the branch of b=1.5 is given a higher preference (S-C2 is assigned with a preference of 0.6), with no preference to S-Cs associated with b=1, a new front is surviving along with a front similar to the initial one (see Fig. 13).

Fig. 10 S-C1 weight=0.6

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common test functions, as well as the consideration of crowding in the context of concepts. Future work of assessing the current algorithm would require not only the existence of an alternative algorithm but also the choosing of an appropriate test suit for concept-based MOP. Moreover the algorithm should be tested with problems concerning more than two objectives. The relation between the number of concepts and the size of the population has to also be investigated. Another possible numerical issue for future investigation is the re-ranking procedure that apart from a through investigation of its algorithm parameters needs to be investigated with respect to the ranks' bounds. These bounds influence directly the fronts that are introduced to the decision makers.

Fig. 13 S-C2 weight=0.6

This can be seen by comparing Fig. 13 with Fig. 10. Any concept can be retained in the evolution by increasing the preferences of its’ relevant S-Cs’, while decreasing the other S-Cs of the problem. For example, concept #6, which has not survived so far, can be elevated by a subjective decision. This can be done by assigning preference weights of 0.7 to S-C8 and to S-C10 as well as assigning S-C1, S-C7 S-C9 a preference weight of -0.4. The resulting front of these assignments, which contains concept # 6 alone, is depicted in Fig. 14.

V. SUMMARY AND CONCLUSION The interactive concept-based search, using MOEA, strengthens symbiosis between computers and humans in exploring conceptual solutions to multi-objective search problems. The algorithm, which is detailed here, allows simultaneous evolution of concepts. The methodology uses ranked non-dominated sets, which are obtained by objective model-based evaluations. These are further manipulated by subjective evaluations to produce an objective-subjective front. The method is extended here to include concepts, which are represented by a tree structure of sub-concepts, with hierarchical preferences. The numerical study, which is detailed in section IV, is followed by suggestions on possible future modifications and developments. ACKNOWLEDGMENT This work has been partially carried out during a Sabbatical stay of the second author with Xin Yao, and the School of Computer Science, at Birmingham University, UK. They should be acknowledged for the support and inspiring environment provided to the second author.

Fig. 14 Concept elevation

REFERENCES

C. Discussion of Some Numerical Issues The non-hierarchical version of the suggested algorithm has been proven to be successful in examples concerning several application areas, including path planning, [2], and mechatronic design (to be published). The results of the academic hierarchical example, which are shown in the previous sub-section, demonstrate the possible extension of the algorithm to other examples and application areas. Regardless of their current success, both the nonhierarchical and the presented hierarchical versions of the algorithm should be carefully studied, and possibly improved, before their general use. The investigation should study the effects of the different parameters, which take a part in the algorithm as detailed in the appendix. A possible improvement might be to tune the penalties with the generation, and in particular a decay of the in-front concept sharing should be considered Another possible approach, which is currently explored by the authors, is to first develop a new algorithm for concept-based MOP (without interactivity or hierarchy), based on NSGA-II ([13]). Such an approach includes elitism, which is known to be significant to MOEA success in solving

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[13]

[14]

I.C. Parmee, “Poor-Definition, Uncertainty, and Human FactorsSatisfying Multiple Objectives in Real-World Decision-Making Environments”, E. Zitler et al. (Eds.): EMO, LNCS 1993, pp: 52-66, 2001. I.C. Parmee, “Human Centric Intelligent Systems for Design Exploration and Knowledge Discovery”. Proceedings of ASCE 2005 International Conference on Computing in Civil Engineering, Cancun, Mexico; July, 2005. G. Avigad, A, Moshaiov, and Brauner N. "MOEA-based Approach to Delayed Decisions for Robust Conceptual Design". In Applications of Evolutionary Computation, Lecture Notes in Computer Science, LCNS 3449, pp: 584-589, Springer, 2005. G. Avigad, A, Moshaiov, and Brauner N, MOEA for Concept Robustness to Variability and Uncertainty of Market’s demands, In the proceedings of the first EC workshop within the 9-th AI*IA conference on AI., September 20-23, 2005 Milan Italy. K. Deb, S. Agrawal, A. Pratap, and Meyarivan, T. A Fast Elitist Nondominated Sorting Genetic Algorithm for Multi-objective Optimization: NSGA-II, In: Proceedings of the Parallel Problem Solving from Nature VI Conference, Paris, France, pp. 849–858, 2000. N. Srinivas, and K. Deb, K., “Multi-Objective Function Optimization using Non-Dominated Sorting Genetic Algorithms”, Evolutionary Computation 2(3), pp: 221- 248, 1994.

within each rank. A sharing penalty function for the i-th C-I, belonging to the m-th concept, within the r-th rank is defined as:

φ ir , m = where,

n r, m

d

r ,m ij

=

no

∑

n =1

⎛ f i n − f jn ⎜ n n ⎜ f best − f worst ⎝

⎞ ⎟ ⎟ ⎠

2

(A6)

where, no is the number of objectives to be optimized, and

f i n is the performance of the particular solution of the i-th individual with respect to the n-th objective. n n n n Also, f best and , are the = max( f i ) f worst = min( f i ) best and worst performances within objective n, of the individuals, in rank r and concept m. Following [14], a sharing function, for the i-th individual with respect to the j-th individual, of the r-th rank and the m-th concept, is computed as:

(A1)

(A2)

shdr,rm,m =

Similarly, a lower (L) bound is assigned for each rank according to:

ij

1−(dijr,m / σ share)2 , if dijr,m ≤ σ share 0, otherwise

(A7)

0 .5 p q

(A8)

where,

(A3)

σ where ε fit1U − 0.5 δ0 r ; 1,..,P Then

rnew = r fiti = fit1U − Δ(rnew −1)genα End For all individuals not assigned with new rank fiti = 0

End In this sorting algorithm Δ is a predefined fitness interval, ‘gen’ is the current generation number, and α is a generationbased search pressure parameter. As α increases, the search pressure increases with generation progress. This is due to the increased span between the fitness of individuals belonging to adjacent ranks as evolution moves from one generation to the next.

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