Artificial Intelligence and Knowledge Engineering

DYNAMIC FUZZY EXPERT MAPS: IDEA AND IMPLEMENTATION Raimundas Jasinevicius, Radvile Krusinskiene, Vytautas Petrauskas, Alelksej Tkaciov Kaunas University of Technology, Department of Informatics, Studentu 50-204A, Kaunas, Lithuania, [email protected], [email protected], [email protected], [email protected] Abstract. Fuzzy Expert Maps (FEM), according to the idea, proposed and developed by the research group of the Computer Literacy Centre (CLC) of the Department of Informatics at the Kaunas University of Technology, is an extension of a very well known Fuzzy Cognitive Maps’ (FCM) concept, preliminary mainly developed at the USC (University of Southern California) by B. Kosko. An essence of the FEMs’ is following: all nodes of an ordinary simple FCM are substituted by fuzzy expert evaluators (some sort of open loop fuzzy controllers), which are able to perform certain decision acts, based on experts' knowledge, represented by lists of rules. Up till now FEMs were dedicated for description of static mutual relationship of entities (nodes) under consideration. The novelty of a theoretical approach, developed in this presentation, is a possibility to include a time dimension into FEM’s description. And this enables researchers to evaluate dynamical changes and time dependent processes during the simulation of FEMs. A practical value of the research in general consists of two parts: 1) a user friendly software package is developed for description of a phenomenon under consideration and for very convenient inclusion of experts’ knowledge for simulation of processes in FEMs; 2) the software package and its supporting materials are delivered to the Fuzzy Engineering laboratory of the CLC for young researchers in the field of soft computing applications under COST IC0702 action. Keywords: fuzzy cognitive maps, fuzzy expert maps, dynamics, expert knowledge, software tool.

1

Introduction: historical discourse

The cognitive maps approach to decision processes’ analysis was started by R. Axelrod at Princeton University ([1]). But it is widely recognized that only after famous L.A. Zadeh’s papers ([2]-[4]) the contemporary avalanche of fuzzy sets applications has burst. Fuzzy control systems (FCS) in industrial applications and fuzzy cognitive maps (FCM) for decision making are the best confirmation of this tendency. By the way, the background for fuzzy thinking and for fuzzy cognitive maps’ applications to decision making processes was preliminary mainly developed at the USC (University of Southern California) by B. Kosko ([5][7]) and later extended by J.P. Carvalho, J.A. Tome, M.S. Kahn, G. Xirogiannis ([8]-[12]) and many, many others. Following the world wide experience in soft computing (for example, [13]-[15]) as well as requirements, emphasized by different decision makers, looking for efficient computerized advisers in various cases of very sophisticated and sensitive situations like financial risk management, medical diagnostics, politics and international relations, environmental protection, terrorism and security, pattern recognition and so on ([16][24]), we have summarized main theoretical features, properties and limitations, used, implemented and/or inherent for FCM-principles-based decision making support tools. The main ideas, captured from the references mentioned above and some theoretical ones developed by the authors of this presentation, were implemented in different decision makers’ support tools at the Computer Literacy Centre (CLC) of the Department of Informatics at the Kaunas University of Technology ([25]-[32]). The most thorough description of the FCMs itself phenomenon such is presented in [7]. There the structure has bivalent nodes (i = 1, 2, …, j, k, … , N) representing fuzzy events, which can be evaluated by concept values Ci in {0, 1}. The nodes are connected by causal edges. The strength of any particular edge, connecting Ci and Ck is represented by wik which takes trivalent value in {-1, 0, 1}. The positive value corresponds to the phenomenon when causal event, actor, goal, trend or concept Ci stimulates the causal event Ck; the negative one corresponds to the effect of suppression and zero demonstrates total indifference. So the connection matrix E lists the causal links between nodes as follows: E = W × C; W = (wik )-matrix, C = (Ci ) – matrix – tulip. In the i-th row the edges wik (k = 1, … , N) are collected from causal concept Ci to causal concept Ck as well as in the k-th column the edges wik (i = 1, … , N) from the concept Ck to the concept Ci are listed. A causal dynamism and conceptual interaction is expressed by the FCM when the concept value Ci on the certain step n of behaviour is subjected to the nonlinear transformation. Such a system can show all possible aspects of interaction between causal concepts Ci (i = 1, …, N) including hidden fixed point, limit cycle or chaotic attractors, corresponding to the real life to be modelled by a simple FCM [7]. - 17 -

The real life is much more complicated. Usually neither concepts are bivalent nor edges are trivalent. In general, Ci are real numbers from the interval [0, 1] and values of edges wik as well are from interval [-1, +1]. So, an ordinary fuzzy cognitive map is described by the following formula: N ⎫ α i [n ] = ∑ w ij C j [n − 1]⎪ j=1 ⎬ ⎪ C i [n ] = Ψi (α i [n ]) ⎭

(1)

This formula corresponds to the structure shown in Fig. 1. Here Ψi (*) are different nonlinearities for all i = 1, …, N, corresponding to the physical meaning of the i-th entity [7].

Figure 1. Structure of the FCM

Nevertheless today's experience permits us to extend FCM nodes concept, including additional fuzzy expert knowledge and enriching representation of real situations under consideration. Such an approach, based on the authors’ research of the possible FCMs’ extension ideas and their transformation into rule-based fuzzy expert maps (FEM) is presented in [28] and [30]. Usually fuzzy reasoning and rule-based inference are inherent in decision making processes ([8], [13], [16]). That is the reason why we were looking for another more adequate mechanism of expert knowledge representation in the tools for decision makers.

Figure 2. Structure of the FEM’s node

In general it is easy to notice that in decision making processes whole fuzzy information to be processed consists of: a) structurally described interactions between entities under consideration, b) some expert characteristics describing the behavior of entities and c) fuzzily prescribed evaluations of strength for each influence. Actually only the first (a) item is given; the second and the third (b, c) ones are guessed in minds of experts, are based on their experience, and are described by a set of fuzzy rules and by the fuzzy logic of their composition ([13]-[16]). In such a case the role of FCM’s node changes significantly. It becomes similar to a - 18 -

fuzzy controller widely described in [14], or [15] and shown in Figure 2 for the node or primary decision maker DJ. In this case each node of FEM contains elements which are inherent for an open loop fuzzy control systems: a fuzzification block (F), a block for fuzzy inference (I) and a block of defuzzification (D). Must be emphasized that fuzzification takes part separately on each input value (E1-EK and D1-DL). The fuzzy inference as well as defuzzification are performed according to the chosen fuzzy logic (for example, – logic operation & (fuzzy min) for rules’ aggregation, logic operation OR (fuzzy max) for Mamdani-type, or Larson’s-product-type output composition and center of gravity (CoG) method for concrete evaluation of decision’s DJ strength) as it is recommended in [7], [13]-[15] and was implemented in [28] and [30]. Up till now FEMs were dedicated for description of static mutual relationship of entities (nodes) under consideration. The novelty of a theoretical approach, developed in this presentation, is a possibility to include a time dimension into FEM’s description. And this enables researchers to evaluate dynamical changes and time dependent processes during the simulation of FEMs.

2

Description of Dynamics: Delay and Inertia Phenomenon in FEM Approach

Dynamics of the processes taking place in the FEM-based simulation of real life activities must be involved: a) by using so-called delay operation (DOP) in each node of the FEM, and b) by changing (slowing down and modelling the phenomenon of inertia) the frequency of calculation steps (steps’ operation –SOP) in each node of the FEM under consideration. Main authors’ efforts in this presentation were directed to elaborate new functions of the proposed FEM node, to introduce more sophisticated inference logic for time-based phenomena. As it is well known, the types of rules of inference in each node of FEM are based on two types of reasoning, which are presented here in the most general form [13]: The first type of rule: IF x is A AND y is B THEN z is C (for Mamdani fuzzy models) (2) The second type of rule: IF x is A AND y is B THEN z = F(x, y) (for Takagi–Sugeno fuzzy models) (3) Defuzzification procedures for the two cases mentioned above can be described as reasoning on the basis of a set of consequents C using the centre of gravity (CoG) or a mean of maximum (MoM) methods for Mamdani type models, and a fuzzy mean (FM) method as reasoning by evaluation of all results z included and processed according to the certain formula Φ(z) for Takagi–Sugeno models ([7],[13]) So, the involvement of DOP (case a)) for a FEM can be achieved by introducing the number of delay steps dij in (2) and (3): (4) IF x[n-1- dij] is A AND y[n-1- dik] is B THEN z[n] is C (for Mamdani fuzzy models) IF x[n-1- dij] is A AND y[n-1- dik] is B THEN z[n] = F(x[n-1- dij], y[n-1- dik] ) (for Takagi–Sugeno fuzzy models) (5) An involvement of SOP (case b)) for the FEM can be achieved from the (2) and (3) according to the (6) and (7), where ki means slowing down ratio and mi – the new step number for the i-th node:

[

]

IF x (m i − 1) − d ij is A AND y

models)

[

[

]

[(m i − 1) − d ik ] is B THEN z [n] is C [

]

[]

(for Mamdani fuzzy

[

]

IF x (m i − 1) − d ij is A AND y (m i − 1) − d ik is B THEN z n = F(x (m i − 1) − d ij ,

]

y (m i − 1) − d ik )

(for Takagi–Sugeno fuzzy models)

(6)

(7)

In both cases mi = ] n/ki [ = 1, 2, 3, … - are strictly integers.

3

Dynamic FEM simulation tool‘s implementation

A practical value of the research in general consists of two parts: 1) a user friendly software package is developed for description of a phenomenon under consideration and for very convenient inclusion of experts’ knowledge for simulation of processes in FEMs [33]; 2) a user’s manual is published and delivered to the Fuzzy Engineering laboratory of the CLC for young researchers in the field of soft computing applications under COST IC0702 action [34]. General views of the FEM dynamics simulation tool’s screens during different stages of simulation are shown in figures 3 and 4 (comments in Lithuanian in the future to be adapted to English language user’s needs).

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Figure 3. View of the FEM dynamics simulation tool’s screen

Figure 4. View of the FEM dynamics simulation tool’s screen

For development of the FEM software graphical tool the Delphi object-oriented programming language was selected. The user's graphical interface is designed, and written using the company’s "Embarcadero" designing tool "CodeGear - Delphi 2009 Architect". Although a market to-day offers a lot of different graphical programs for the same functions, the Microsoft Paint and Adobe PhotoShop software were selected because of there fast performance. User's Manual (Guide) has a web-based structure and uses Adobe Dreamweaver program. The first examples of a practical use of the tool under consideration are delivered in [28], [35] and [36]. The [28] is devoted to the problem from a field of international relations: a determination of an optimal tine for the USA troops withdrawal from Iraq when a time –delay phenomenon must be taken into account during the FEM simulation. The [35] and [36] describe a medical diagnostics problem formulation based on Takagi-Sugeno type FEM used for pattern recognition.

4

Concluding remarks and acknowledgements

This article has targeted first of all to present a historical discourse of developments and extensions of fuzzy cognitive maps’ concept and to show how inclusion of more sophisticated knowledge into this concept permits to invent fuzzy expert maps. Another target of this presentation is a demonstration of a very desirable possibility to simulate dynamic properties of those fuzzy expert maps by including time dimension and time delays in each node. The third goal of the article is a some sort of advertising of a new user friendly software tool for decision makers. Must be emphasized that restrictions on the size of the article have not permitted to - 20 -

demonstrate any practical implementations and use of the new tool, and the next article of our team will be devoted for presentation of practical examples of the tool’s use by decision makers. Authors are eager to express their gratitude to young researches who took part in tools’ development processes and to acknowledge the synergetic influence on theses considerations made by the COST program Action IC0702 “Combining Soft Computing Techniques and Statistical Methods to Improve Data Analysis Solutions” under coordination of prof. Christian Borgelt from European Centre for Soft Computing (Mieres, Spain).

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AN INVESTOR RISK PROFILING USING FUZZY LOGIC-BASED APPROACH IN MULTI-AGENTS DECISION SUPPORT SYSTEM Andrius Jurgutis1,2, Rimvydas Simutis1 1

Kaunas University of Technology, Department of Process Control, Studentu str. 50-163,Kaunas, Lithuania, [email protected], [email protected] 2 Bank “Finasta”, Maironio str. 11, Vilnius, Lithuania Abstract. The paper present the fuzzy logic expert system called MADSYS for an investor risk profiling and the first results of this test system. MADSYS system will be used in the interface agent (agents) of multi-agent investment management information system. One of the principal tasks of the multi-agent system is to help an investor to make investment decisions and to provide appropriate investment proposals according to the investor’s risk profile. From MADSYS depends a lot of things, namely the multi-agent investment management information system accuracy, proposed investment decisions, reliability, investor satisfaction. The usage of MADSYS system in the multi-agent system makes it more intellectual, i.e. the system will be able to adjust automatically to the changing of investor’s risk tolerance. The MADSYS system may be tried online at the following address: www.sprendimutechnologijos.lt/webapp. Keywords: multi-agents decision support system, risk profiling, fuzzy logic.

1

Introduction

Our works and tests are directed to formation of intellectual information system for investment management. As we know, the investments management is a complicated and complex task that converges into settlement of several parallel and consistent tasks (performance of tasks) and control and management of separate tasks [9], [14], [22]. For example, when the stocks trading moved to the electronic space, the financial markets became global and dynamic, i.e. it is easy for the investor to make transactions in any world market, to convey and receive information, whereas on the other side, in order to invest successfully, it is necessary to comprehend quickly the subtleties of new financial markets, to follow the information continuously, to be able to evaluate it competently, and to make the investment decisions quickly. It is doubtful that one person can do this, whereas the team work possibly is not effective economically and cannot compete with new investment management information technologies. We think that the software agents are the most suitable to create the desired system. The multi-agent system that they form is able to perform parallel and consistent tasks of investment management. There are some examples of such systems, e.g., WARREN, MASST [23]. Thus it is possible to imagine the multi-agent investment management information system that we construct as the autonomous virtual investment management and consultation company, where the intellectual software agents perform the work done by specialists in the classical company of investment management, i.e. software agents replace economic advisers, statisticians, financial brokers, analysts, etc. Briefly, the multi-agent information system has to make suggestions to the investor, when and which securities should be bought/hold/sold with regard to the situation in financial markets and investor’s profile, and in such a way to manage successfully the investment portfolio. The authors of the books and articles on multi-agent information systems for investment management state numerous advantages of these systems in various aspects, whether these were economic, technological or other aspects [9], [24]. In this article we already do not present detailed information that we have presented in previous publications [9], [10] about the advantages and structure of the multi-agent investment management information system that we are creating, connections between agents and working mechanism. This is not important to MADSYS system presentation. In this article we will focus on the fuzzy logic-based MADSYS system used in the interface agent, which main task is to assess the investor’s risk tolerance (profile) as accurately as possible. The MADSYS system in the interface agent is very important for the entire multi-agent information system, because its proper operation determines qualitative work of other agents, precision and reliability of the multiagent system, and quality of satisfaction of the investor’s needs. As we have reviewed briefly in the introduction the presumptions of creation of multi-agent information system in the second part we will introduce, how the MADSYS system is going to assess the investor’s risk tolerance. The third part of the article will introduce how the fuzzy logic helps to asset the investor’s risk tolerance in the MADSYS system. In the fourth part of the article we will present the testing results of MADSYS system, and finally we will use conclusions to present the generalizations and achieved results.

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2

The risk tolerance assessment methods

The success of the multi-agent investment management information system is essentially hidden in the as much precise assessment of the investor’s risk tolerance as possible. The interface agent transfers this parameter to other agents of the system, which goal corresponds to this value, i.e. information on the riskiness of the securities being searched for [9]. The improperly determined investor’s risk tolerance may result in faulty suggestions of the system to the investor, i.e. the investor will be offered improper securities, which will result in non-tolerated losses or non-receiving of desired profit, thus he will stay unsatisfied with the investment results, will not trust the system and finally stop using it. It is a pity but it is difficult to measure the risk tolerance. This task has been discussed among various scientists for many years already, for example among the psychologists, who divide the human behaviour in two groups: intellectual behaviour and emotional behaviour (spontaneous, unpredictable). Unfortunately the risk tolerance is attributed to the area of emotional behaviour [16], which makes it even more difficult to measure. It is also discussed by economic analyzers [7], financial engineers and creators of artificial intelligence [1], [19]. The only correct and precise method to measure the risk tolerance has not been found yet [18]. It is possible to measure the investor’s risk tolerance in three modes: from the collected autobiography of investor, report on communication with the investor, and from special questionnaire [4]. In our case we favoured special questionnaire meant to measure risk tolerance. It would seem that everything is clear and simple, but a lot of questions arise how to make “good” questionnaire [10]. For example, how many questions should make the questionnaire? [17], what should the questions be about? [11], [21], how long should it take to answer the questionnaire? how to assess the precision of questionnaire?, and other questions [10]. The determination of risk tolerance is difficult task and it is also reflected in the published scientific researches, which conclusions state that the client evaluates the risk tolerance himself the best if compared to the ability of financial advisers to determine it using the questionnaire. The correlation coefficient between these evaluations reaches only 0,4 [17]. Hanna and Roszkowski have worked a lot with formation of questionnaires and evaluation of precision. They suggested that the questionnaire on tolerance to risk should consist of various questions, i.e. questions about selection of investment alternatives [6], [8], questions about combinations of financial means in portfolio, and subjective questions [5], [12], questions assessing current behaviour of the investor [15], and hypothetical questions with clearly described scenarios of financial market. Several methods were suggested to assess the reliability and accuracy of questionnaire about risk tolerance. One of them is based on the correlation value between self-evaluation (independent) and risk tolerance determined by financial agent. The second method is based on measurement and evaluation of standard error of risk tolerance [2], [3]. The third method is based on the evaluation of correlation between the pairs of answered questions in the questionnaire. To summarize this section, we would like to share some practical observations. The analysis of risk tolerance questionnaires showed that the majority of them are directed to asset allocation in portfolio and trading strategy identification. When an investor answers the questions at the end of the questionnaire he can find portfolio structure according to asset classes, whereas a financial intermediary cannot know his client and identify investor risk tolerance and is danger of losing a client in future. We will try to avoid this in creation of our system. Besides, we would like to remark that the questionnaire consists of the questions with clearly determined variants of usually 3-5 answers, each of which has certain value in points, while the final conclusion is made after the sum of collected points is assessed. Besides, usually one answer is allowed, although you would agree that there are cases when you want to insert “own” answer between the presented ones. In our opinion, this harms the precision of questionnaire because the limited number of the answers in advance already determines, to which risk group the investor will belong, i.e. when the investor chooses one of the limited answers in the questionnaire, he attributes himself to some risk class, in other words, the answer’s variant is closely related to certain risk group. While creating our system we suggest rejecting close (determined) relation between the answer and risk group. Therefore we suggest using different principle (structure) of question formation that is not to make the determined answers to the question – the investor should express answers freely. We consider that it would be innovative to include certain statements about investor or investments into the questionnaire, whereas the investor would answer in the scale from 0 to 100%, how the statement matches to him, what degree of the offer he would use, etc. In such a way we would allow the investor presenting “own” real answer, or we would introduce “fuzziness” in the terms of fuzzy logic sets. Completing this section we will make several conclusions, which we use to create our evaluation method of risk tolerance. First of all, we will use the questionnaire to measure the tolerance to risk. Secondly, we will avoid “bad” questions while making the questionnaire [10]. Third, we will take into account the conclusions and recommendations of published scientific works and make our questionnaire out of 17 questions-statements (15 questions, 2 statements). Fourth, as many unknowns remain in present evaluation of risk tolerance, how to do it precisely, and in order to grant the investor possibility to submit “own” answer in the questionnaire, we think that it is meaningful to measure the investor’s risk tolerance using fuzzy logic. Further in the article we will review more widely the suggested fuzzy logic system for investor’s risk tolerance assessment. - 24 -

3

Fuzzy logic-based investor‘s risk profiling system MADSYS

We know from the theory that the construction of expert fuzzy logic system consists of four steps: introduction of fuzziness, formation of expert rules, aggregation, and elimination of fuzziness (defuzzification) [13]. We will discuss all the steps in more details. We will consider that the reader of the article has experience with the systems of fuzzy logic, thus we will not sometimes maintain the consistency of steps in order to present and explain easily the working of MADSYS. We will start from the structure of MADSYS system expressed in the Fig. 1. As you see, the MADSYS system consists of 15 simple (two input parameters and one output) Mamdani fuzzy logic systems arranged in certain order, which we have called cascade arrangement (cascade structure) or cascade system of fuzzy logic. We have selected such structure in order to reduce the number of expert rules used in MADSYS system. As we know from the theory of fuzzy logic [20], it is not difficult to calculate that in presence of 16 input parameters if one from some five values is attributed to each parameter, the total needed number of rules would be very big in order to make the final conclusion, i.e. several hundred million rules. Therefore in our case the simple fuzzy logic system consisting of 16 input parameters (answers to the questions of the questionnaire) and one output (evaluation of tolerance to risk having five options) will be ineffective, clumsy, difficult to correct and adjust. The MADSYS structure provided in the Fig. 1 allows reducing the number of rules down to 375 rules. We can say that some rules will repeat, which means that the number of unique (not repeated) rules will be smaller. Such structure makes MADSYS flexible, easy and quickly to correct. MADSYS system Q2

Q3

Q4

F1

Q5

Q6

F2

Q7

Q8

F3

Q10

F4

O1 O2

O3 O4

F9

F10 O9

Q9

Q11

Q12

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Q14

F6

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Q17 F8

O7 O8

F11

F12 O11

O13

Q16

F7

O5 O6

O10 F13

Q13

O14

O12 F14

F15 O15 (Output: client risk tolerance) Figure 1. MADSYS structure

According to Fig. 1, first eight Mamdani fuzzy logic systems (from F1 to F8) are “fed” from the questionnaire. Besides, we divided the questions into four groups, i.e. general question, hypothetic questions, experience questions and risk questions. As you can see, this division is important for the system structure and for the final conclusion formation. Each group consists of four questions-statements, which are numbered accordingly from Q2 to Q17. The first question Q1 is used to ask the investor, to which risk groups from present five (especially low risk tolerance, low risk tolerance, moderate risk tolerance, high risk tolerance, especially high risk tolerance) he attributes himself. The evaluations of risk tolerance received later by MADSYS will be compared to the answers of investors Q1 and the conclusions will be made about accuracy of the MADSYS system using the methods discussed in the second section of this article. Other simple Mamdani fuzzy logic systems (from F9 to F15) are “fed” from the results aggregated by previous pairs of fuzzy logic systems after elimination of fuzziness (but without conversion to the linguistic value; hereinafter fuzzy value), as it is shown in Fig. 1. The defuzzification in the fuzzy logic systems from F1 to F14 was using the LOM method (largest (absolute) value of maximum), whereas the method of “centre” was used in the last F15. We have selected such defuzzification because when the cascade is going downwards, the simple fuzzy logic systems already on the third layer converge towards the “centre”, i.e. the MADSYS system would reach the same conclusion at any answers of the questionnaire that the level of investor’s risk tolerance is moderate. It is evident that it is not good, thus we selected the aforementioned elimination methods of fuzziness and sequence that solves the problem. As we have already mentioned, the questions were grouped to four groups. According to Fig. 1, at the beginning the conclusions about investor’s risk tolerance are made from the pairs of questions, while later (second layer) they are made from the group questions (the groups in the Fig. 1 are distinguished in different background: rarely spotted background – general questions, densely spotted – hypothetical questions, vertical lines – experience questions, horizontal lines – risk questions). Later the conclusions are made from the groups - 25 -

of neighbouring questions. The groups of questions are arranged with certain logic and insight, more particularly the input parameters to the last fuzzy logic system F15 would the conclusions received from the generalhypothetic and experience-risk questions. In such a way one more advantage of cascade structure is highlighted – the conditions are created to make more accurate conclusion about the investor’s risk tolerance. The conclusion received from the experience-risk questions would determine more the “push” of the final conclusion towards more extreme evaluation of risk tolerance, whereas the conclusion received from general-hypothetical questions would determine more the “centre” or “average” of the final conclusion. We formed the expert rules on the basis of our experience exceeding nine years in consulting of investors, providing the recommendations to them and using general recommendation from financial engineering literature [1], [13]. As the MADSYS system consists of simple Mamdani fuzzy logic systems with two input parameters and one output parameter, the expert rules of system shall be written in the simple generalized form expressed by Formula 1. IF Ii is Xi AND Ij is Xj

THEN O is X

(1)

where I – input parameters (in fuzziness value) in simple fuzzy logic systems (value in the scale from 0 to 100), X – the value of one from five possible groups of risk tolerance (linguistic variables: especially low risk tolerance (RL0), low risk tolerance (RL1), moderate risk tolerance (RL2), high risk tolerance (RL3), and especially high risk tolerance (RL4)). Finally it is time to discuss about the membership functions used in the simple Mamdani fuzzy logic systems (they are marked with symbol µ in the fuzzy logic). They are illustrated in Fig 2. Triangular membership functions Input membership function

Trapezoidal membership function Input membership function

Output membership function

Output membership function

a) Input parameter received from general-hypothetical questions

c) Input parameter received from general-hypothetical questions

Input parameter received from experience-risk questions

Input parameter received from experience-risk questions

Output membership function

Output membership function

b)

d) Figure 2. Used membership function in MADSYS system

Membership functions links the fuzzy questionnaire answers and F9, ... ,F15 input parameters with the linguistic values of risk tolerance levels, as well as the meanings of all fifteen conclusions of simple fuzzy logic systems with linguistic values of risk tolerance. While testing the MADSYS system, we used two types of membership function, i.e. in one case the triangular types (were used everywhere) and in the other case – trapezoidal. We used the cascade structure of MADSYS system and tried to maintain the simplicity of the - 26 -

system, its quick and easy calibration, thus we used the repeated elements for testing. Simple fuzzy logic systems from F1 to F14 have the same membership function of input and output parameters, which are shown in Fig. 2 a) triangular membership functions, and Fig. 2 c) trapezoidal membership functions. However, the last F15 system has different membership function just for input parameters, which are shown in Fig. 2 b) as triangular membership functions and in Fig. 2 d) as trapezoidal membership functions. We do so to have the conclusion (as it has been already mentioned) received from the experience-risk questions to determine more the “push” of the final conclusion towards more extreme evaluation of risk tolerance, while the conclusion from generalhypothetic questions would determine more the “centre” of the final conclusion, and thus the membership function of input parameters were corrected accordingly. To make the system simple, all the expert rules used in the simple fuzzy logic systems are similar, which means that MADSYS uses only 25 simple rules written in the Formula 1 form. The testing results of MADSYS system with the types of one or another membership functions are presented in the fourth section of the article. However, when the system was tested the idea rouse to try simplifying the MADSYS structure even more by using collected questionnaires (data for training), and ANFIS abilities. The modified structure of MADSYS is presented in Fig. 3. It shows that the cascade structure of the system gets shorter (less by one layer or less by two simple fuzzy logic systems), while the last fuzzy logic system “FS” is not of Mamdani type, but of Sugeno. As it is known, the Matlab “Fuzzy Toolbox” ANFIS system works with the fuzzy logic systems of Sugeno type [20]. In such a way we can use the ANFIS system to generate the fuzzy logic systems of Sugeno type with four input parameters (inputs are conclusions from four question groups of fuzzy value) and one output parameter, i.e. size of risk tolerance. The testing results of the modified MADSYS system are presented in the fourth section of the article. MADSYS modified system Q2

Q3

Q4

F1

Q5 F2

O1 O2 F9

O9

Q6

Q7

Q8

F3

Q9

Q10

F4

Q11

Q12

F5

F6

O3 O4

O5 O6

F10

F11 O10

Q13

O11

Q14

Q15

Q16

F7

F8 O7

O12

Q17

O8 F12

FS OFS (Output: client risk tolerance) Figure 3. Structure of modified MADSYS system

4

Testing results of MADSYS system

Before we present the testing results of MADSYS system, we will introduce briefly the testing environment. The visitors of the website “Traders.Lt” (www.traders.lt) that connects investors and they operates in Lithuania were invited to test the system. The short publication about MADSYS was published on that website and it was suggested to try it for free. Over 900 visitors of the website have read the publication in the period of one month, but only 143 visitors connected to the system and tried it. The time was measured that takes one visitor to answer the questions in the questionnaire of the MADSYS system. On average it used to take 6 minutes to the investors. In order to assess the accuracy of MADSYS system more precisely and objectively, we did not use the answers of the investors, who had answered the questionnaire in less than 2 minutes in the statistical calculations. We think that such answers have no value, because probably the answers were selected accidentally, without getting deeper in the question, and such answers can distort the reliability evaluations of the system, and thus they cannot be used to adjust the system. Therefore only 96 questionnaires were left, which were suitable to evaluate the MADSYS system and to adjust it. They were used to get the testing results of MADSYS system, which are discussed and presented below. First of all we calculated and assessed the correlation coefficients between self-evaluation (independent) of the investor (Q1) and evaluation of risk tolerance presented by MADSYS system (O15), and correlation coefficients between interim conclusions (see Fig. 1 O9, O10, O11, O12, O13, O14). As it has already been mentioned, we did testing using two different membership function type, i.e. triangular and trapezoidal. Besides, in order to assess the system more extensively, we have distinguished two strategies of correlation calculation, - 27 -

i.e. correlation coefficients were calculated using fuzzy values (in scale 0-100), and expressive values (linguistic values encoded by numbers from 1-especially low risk to 5-especially high risk). The corresponding matrixes of correlations are presented in Table 1 (in case of triangular membership function) and in Table 2 (in case of trapezoidal membership function). Table 1. Correlation matrixes when membership functions is triangular Q1

O9

O10

Q1 O9 O10 O11

1 0,356309 0,269305 0,465042

1 0,413866 0,638101

1 0,463784

1

O12

0,378204

0,622825

0,542866

0,581922

1

O13 O14

0,351798 0,414051

0,711731 0,661274

0,853189 0,480698

0,589446 0,84775

0,597509 0,814971

1 0,612566

1

0,624217

0,757567

0,770646

0,817952

0,881242

O10

O11

O12

O13

O14

1 0,437579 0,495374 0,870627 0,423951

1 0,530727 0,568728 0,822092

1 0,578228 0,816348

1 0,582046

1

0,697862

0,736453

0,654049

0,833463

O13

O14

O15 0,410808 0,703402 Correlation matrix using fuzzy values Q1 O9 Q1 1 O9 0,44732 1 O10 0,281569 0,451918 O11 0,519122 0,575439 O12 0,376108 0,638895 O13 0,386569 0,720073 O14 0,427973 0,65368

O15 0,313077 0,632733 0,457035 Correlation matrix using expressive values

O11

O12

O13

O14

O15

1 O15

1

Table 2. Correlation matrixes when the membership function is trapezoidal Q1

O9

O10

Q1 O9 O10 O11

1 0,313698 0,163745 0,485225

1 0,173646 0,424472

1 0,285712

1

O12

0,395987

0,515933

0,493271

0,507683

1

O13 O14

0,230755 0,508173

0,637644 0,411168

0,692843 0,302488

0,220676 0,844005

0,439766 0,704369

1 0,23818

1

0,360112

0,801745

0,694502

0,285868

0,966034

O10

O11

O12

O13

O14

1 0,249001 0,422983 0,702807 0,200621

1 0,538157 0,304664 0,810033

1 0,266123 0,732806

1 0,140703

1

0,831122

0,750875

0,241003

0,884853

O15 0,477424 0,449695 Correlation matrix using fuzzy values Q1 O9 Q1 1 O9 0,228246 1 O10 0,100782 0,170601 O11 0,390161 0,429387 O12 0,433592 0,38061 O13 0,058926 0,621437 O14 0,408209 0,292232

O15 0,452093 0,373882 0,29179 Correlation matrix using expressive values

O11

O12

O15

1 O15

1

According to the Table 1, when the triangular membership functions are used, the correlation coefficient between self-evaluation of the investor (Q1) and system’s evaluation (O15) is 0,41, i.e. it corresponds to the number given in the literature [17]. If we used expressive values in simple fuzzy logic systems, which form MADSYS system, we would see as it is evident from Table 1 that the system’s work would not be as reliable, because the correlation coefficient decreases from 0,41 to 0,31. - 28 -

Accordingly Table 2 shows that when the trapezoidal membership functions are used, the correlation coefficient is higher and is equal to 0,47, whereas the usage of expressive values reduce the correlation coefficient down to 0,45 again. According to the data provided in the tables, two conclusions can be made: the system evaluates tolerance to risk more precisely when the trapezoidal membership functions are used, and second, the usage of expressive values in simple fuzzy systems do not provide higher accuracy in the final conclusion. The analysis of correlations shown in Tables 1 and 2 between the group questions (between O9&O10, O11&O12, O9%O15, O10&O15, O11&O15, O12&O15) shows that the correlation coefficients vary a lot and the correlations between the group questions are often very weak. However, the received results in this developmental stage of the system raise more questions than answers. Weak correlations between the group questions and final conclusions show that there is conflict and the question arises: in which place? Is the formulation of questions not suitable? Do the investors “deceive” themselves by answering to the questions? The results that we have received made us reconsidering another tactics, how to select such answers of the investors, which would help to adjust the system. We see two methods to solve this problem. The first one is the one that we are developing – to use two data bases with the answers of investors. One base should store all the answers of the investors, while the other should contain only such answers, which result in conclusion satisfying the investor or if the investor agrees with the conclusion. In such a way the MADSYS system would be adjusted using the data of the second data base, and it would be attempted to select such parameters of the system, which would result in higher correlation coefficients between the group questions and final conclusion. The second method would be to use the data base of experts or “honest” investors answers (standard data base) to adjust the parameters of the MADSYS system. When we analyzed, which MADSYS system (using triangular membership function or using trapezoidal membership function, or modified MADSYS system) is more precise (in the sense of correlation between Q1 and O15), we calculated the average and standard deviation of error provided by the system to the risk tolerance. The received results are presented in Table 3. As we see from the Table 3, the best results again were received using the trapezoidal membership functions. The results of the modified system are bad and they should not be taken in account, because we have too little training data. Table 3. Comparison of error’s average and standard deviation provided in MADSYS system

Average Standard deviation

Using triangular membership function 23

Using trapezoidal membership function 13

Modified MADSYS system using ANFIS 28

12

9

23

Finally the Table 4 presents the calculated distributions, i.e. we calculated, how accurately three tested aforementioned variants of MADSYS system evaluated the risk tolerance compared to the one defined by the investor. In other words, we calculated the percentage of correspondence between the evaluation of MADSYS system and investor (Q1=O15±0), and how much they differed in one risk level (Q1=O15±1), two risk levels (Q1=O15±2), and three risk levels (Q1=O15±3). Table 4. Evaluation of precision of tested MADSYS systems

Q1=O15± 0 1 2 3

Using triangular membership function 15 % 68 % 17 % 0%

Using trapezoidal membership function 37 % 56 % 7% 0%

Modified MADSYS system using ANFIS 18 % 64 % 0% 18 %

According to the Table 4, using the trapezoidal membership functions the MADSYS system achieves the best results, i.e. among 100 investors, the evaluation of risk tolerance O15 of 37 investors corresponds to the investor’s Q1, whereas the difference by only one risk level was noticed in case of 56 investors, which could be regarded as good result. In general, all the tested variants of the system allow evaluating the tolerance to risk correctly (up to mistake of one risk level) in case of 80% of investors, who have answered the questionnaire. Thus we can suggest that the remaining 20% investors, with regard to whom the evaluation of the system differs by more than one risk level, should think, whether they really know themselves with regard to the risk tolerance.

5

Conclusions and further works

This article has presented the investor risk tolerance evaluation system called MADSYS based on the fuzzy logic, which will be important for the multi-agent investment management information system that is - 29 -

being constructed. The reliability and accuracy of all information system will depend on how accurately we evaluate the risk tolerance. As the MADSYS system is formed from simple Mamdani fuzzy logic systems arranged in cascade structure with the repeated number of expert rules, and having assessed the receives testing results, we can say that MADSYS suits well to evaluate the investor’s risk tolerance. After we collect more answers to the questionnaire, we plan to develop better the modified MADSYS system and we expect to make it work better. We have implemented the risk tolerance evaluation system with the help of MATLAB “Fuzzy Toolbox”. The MATLAB package “Builder JA2” allowed implementing the MADSYS system online.

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Bojadziev G., Bojadziev M. Fuzzy Logic for Business, Finance, and Management. 2nd Edition. World Scientific Publishing Co. Pte. Ltd. 2007. Bright J., Adams A. The ProQuest Risk Profiling System. Technical Manual. The University of New South Wales. 2000. Callan V. J., Johnson M. Some Guidelines for Financial Planners in Measuring and Advising Clients about Their Levels of Risk Tolerance. Journal of Personal Finance. 2002. Droms W. G., Strauss S. N. Assessing Risk Tolerance for Asset Allocation. Journal of Financial Planning. 2003. Grable J. E., Lytton R. H. Financial Risk Tolerance Revisited: The Development of a Risk Assessment Instrument. Financial Services Review. 1999, volume 8. Hanna S., Gutter M. A Theory Based Measure of Risk Tolerance. Proceedings of the Academy of Financial Services. 1998. Hanna S. D., Gutter M. S., Fan J. X. A Measure of Risk Tolerance Based on Economic Theory. Journal of Financial Counseling and Planning. 2001, volume 12. Hanna S. D., Chen P. Subjective and Objective Risk Tolerance: Implications for Optimal Portfolios. Journal of Financial Counseling and Planing. 1997, volume 8. Jurgutis A., Simutis R. Building of securities valuation IT system using multi-agents approach. Pproceedings of the 4TH international conference on electrical and control technologies. Kaunas Technologija. 2009.

Jurgutis A., Simutis R. An Investor Risk Tolerance Assessment Using Interface Agent in Multi-agents Decision Support System. Pproceedings of the 16TH international conference on Information and Software Technologies, IT 2010. Kaunas Technologija. 2010. LeBaron D., Farrelly G., Gula S. Facilitating a Dialogue on Risk: A Questionnaire Approach. Journal of Financial Planning. 1989. Mittra S. Practicing Financial Planning: A Complete Guide for Professionals. Mittra & Associate. 1995. Negnevitsky M. Artificial Intelligence: A Guide to Intelligent Systems. 2nd Edition. Addison-Wesley. 2005. Oates T., Nagendra Prasad M. V., Lesser V. R. Cooperative information gathering: A distributed problem solving approach. Technical Report, Department of Computer Science, University of Massachusetts. 1994. Pratt J. W. Risk Aversion in The Small and in The Large. Econometrica. 1964, volume 41. Roszkowski M. J., Davey G., Grable J. E. Insights from Psychology and Psychometrics on Measuring Risk Tolerance. Journal of Financial Planning. 2005. Roszkowski M. J., Grable J. Estimating Risk Tolerance: The Degree of Accuracy and the Paramorphic Representations of the Estimate. Journal of Financial Counseling and Planning. 2005, volume 16. Roszkowski M. J. How to Assess a Client’s Financial Risk Tolerance: The Basics. Personal Financial Risk Tolerance. The American College. 1992. Shanteau J. Expert Judgment and Financial Decision Making. Risk Behaviour and Risk Management. Proceeding of the First International Stockholm Seminar on Risk Behaviour and Risk Management. 1995, pp. 16-32. Sivanandam S. N., Sumathi S., Deepa S. N. Introduction to Fuzzy Logic Using MATLAB. Springer. 2007. Yook K. C., Everett R. Assessing Risk Tolerance: Questioning the Questionnaire Method. Journal of Financial Planing. 2003. Zeng D., Sycara K. Coordination of multiple intelligent software agents. International Journal of Cooperative Information Systems. 1996. Zeng D., Sycara K., Decker K., Williamson M., Pannu A. Distributed intelligent agents. The robotics institute, Carnegie Mellon University. 1996. Wooldridge M. An Introduction to Multiagent systems. John Wiley & Sons, LTD. 2006.

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A FRAMEWORK FOR KNOWLEDGE-BASED CONFIGURATION OF PROJECT MANAGEMENT INFORMATION SYSTEMS Solvita Berzisa, Janis Grabis Riga Technical University, Faculty of Computer Science and Information Technology, 1 Kalku, Riga, Latvia, [email protected], [email protected] Abstract. Project management is a complex process regulated by project management methodologies, standards and other requirements. Project management information systems are used to support project management activities. To ensure effective use of the project management information system, it is necessary that the system could be configured according requirements of the chosen methodology. The configuration provides a structural framework for project management while accumulated project management knowledge provides basis for efficient project execution. Therefore, the project management information systems also should provide means for representing and reusing knowledge. The objective of this paper is to propose architecture of knowledge-based system that ensures project management information system configuration process with appropriated knowledge. This architecture is referred as the knowledge-bases configuration system. Project management knowledge repository and case-based reasoning principles are used in development of the proposed architecture. This architecture allows storing project management and configuration information and organizing and reusing this information. Keywords: PMIS configuration, knowledge-based system, case-based reasoning.

1

Introduction

Establishing project management (PM) procedures and setting up a PM environment are among the main tasks of project initialization [1][2]. PM methodologies are used to support this activity. However, the methodologies are fine-tuned for every particular project to better account for specific features. PM knowledge can be used to assist the fine-tuning. The objective of this paper is to develop an approach for using knowledge in configuring the PM environment. Given that the PM environment usually is implemented using the project management information system (PMIS), the knowledge utilization is proposed a part of the overall configuration of PMIS. The configuration defines data structures and processes supported by PMIS. Consequently, knowledge accumulated during previously completed projects can be used to identify what data structures and processes should be considered in the configuration for the current project. Case-based reasoning (CBR) principles are used for development of the PM knowledge-based configuration system (KBCS). This architecture allows to store theoretical and practical knowledge in the repository and reuse it in PMIS configuration for similar projects. All used description of the PMIS configuration has been added to knowledge repository as a new case. The paper describes the PMIS configuration approach and the PM KBCS architecture. The contribution of this research is development of the PM KBCS architecture. To our knowledge, knowledge processing has been applied to addressing individual project management activities while there are no applications in setting-up the project management environment and in configuration of PMIS in particular. Some PM knowledge management solutions have been developed for software planning [3][4][5] and size estimation [6], and constructing schedule development [7][8]. All above mentioned PM knowledge management solutions have been provided for some PM tasks and commonly have been associated with certain areas. The proposed KBCS architecture allows PM information and PMIS configuration information and knowledge reuse in preparing of description of the PMIS configuration. PM KBCS can be used for projects in different areas assuming that an appropriate methodology is available. The proposed architecture with some modifications also could be used for knowledge representation in PMIS. This paper is divided in six sections. Section 2 is a description of the PMIS configuration approach. Main elements and technologies of the knowledge-based system are described in Section 3. This section also provides a brief comparison of the proposed system with some of existing systems. Structure of PM KBCS architecture is explained in Section 4. An example of knowledge use in PMIS configuration is described in Section 5. Section 6 concludes and discusses future work.

2

PMIS Configuration Approach

It is assumed that a project is performed using a project management methodology and PMIS is used to support project management activities. Before its productive usage, PMIS must be configured according to requirements set by the project management methodology used. The purpose of the PMIS configuration approach elaborated in [9] and shown in Figure 1 is to provide a systematic solution for transforming the informally specified requirements Rj into an executable PMIS configuration Ijk, where j indicates the project and - 31 -

k refers to the particular PMIS. This transformation process consists of two steps. The first step is transformation of Rj into the standardized form Cj, where Cj is structured in a format defined by the template S: Cj = T1 (Rj,S) (1) A configuration client is used to support this manual transformation. The second step is transformation of Cj into an executable configuration of PMIS Ijk : Ijk = T2k (Cj) (2) k One configuration file Cj can be transformed to the various PMIS configuration Ij by using different transformations T2k. The same transformation can be applied to different Cj assuming the same target PMIS. The template S is defined using the XML schema for Configuration of Project Management information system (XCPM) [10]. XCPM allows describing PM data, processes and knowledge. The schema is developed on the basis of the PM concept model that has been obtained after conceptual modelling of the project management domain [11]. The concept model describes the PM domain concepts and it relations. The standard template also includes abstract elements available for representing project features not supported by default. Given the schema provides a comprehensive definition of the project management domain, it can be used to represent various PM methodologies and standards. As the result of transformation T1, the schema is populated with data characterizing particular project and Cj is represented as an XML file. The configuration process is supported by appropriate knowledge from the PM knowledge repository. The PM knowledge repository includes information about PM methodologies, previously used PMIS configurations Cj, j=1,…m, and other relevant information. Two types of knowledge are required during the configuration process: configuration and operational. The configuration knowledge is necessary for T1 and is used in configuration client. Operational knowledge is used for T2k and during usage of PMIS. It includes diagrams, templates, documents, checklists and other.

Figure 1. PMIS configuration approach

3

Knowledge-based Systems

A knowledge-based system (KBS) is the computer system that contains stored knowledge and solves problems like humans would. KBS has been an active research theme for the last twenty years. Consequently, there are a lot of different KBS developed for different fields: medicine, transport, customer service management, logistic, enterprise, finance etc. Knowledge-based configuration is one of the most successful application areas of KBS hence several approaches have been developed to tackle configuration tasks [12]. It is used in a number of domains, such as, telecommunication, engineering, process control, software configuration management, computer design, electronics, chemical design and others. All KBS systems consist of four components: a knowledge base, an inference engine, a knowledge engineering tool, and a specific user interface [13]. The knowledge base stores knowledge and can be implemented as a repository, a relational database or a case library. Knowledge modelling and engineering activities and tools deal with development of the knowledge base. The inference engine is mostly one of reasoning technologies: rule-based, case-based and hybrid. Rule-based reasoning uses ‘if-then-else’ rule statements [13]. Knowledge is represented in rules. New problem is solved by finding appropriate rule in the knowledge database. Rule-based reasoning has been used in some configuration systems [12] and expert systems such as product planning, teaching, system development, knowledge representation and other [13]. Only one of reviewed PM KBS [6] used rule-based reasoning together with data clustering. The most frequently considered knowledge processing method is CBR that uses the past experience [14]. CBR is used in KBS for project management [3][4][5][7] and also in the proposed PM KBCS architecture. Knowledge is represented with the cases. A new problem is solved by finding the similar past cases and re-using in the new problem situation. CBR consists of the case library and four-step process: retrieve, reuse, revise and retain [14]. During the retrieve step, the most similar cases are founded in the case library according the new - 32 -

case description. Information and knowledge from similar cases are copied or adapted to solve the new problem in the reuse step. Proposed problem solution is tested, evaluated and repaired during the revise step. During the retain step, the new case and knowledge is added to the case library for future reuse. Following technologies is used to ensure CBR implementation: data warehouse and online analytical processing (OLAP) [15] [16] [17]; data mining [18][19]. CBR principles have been used in Google translator [20]. Some KBS already use both rule-based reasoning and CBR [21] [22] [23]. Proposed PM KBCS is similar with the recommender system for software project planning [3] and the knowledge-based logistics strategy system [15]. The recommender system for software projects planning uses project attributes and weight to find similar cases and ensures statistical information about cases similarity and performs multiple cases analysis. The proposed PM KBCS also uses the set of project attributes and similarity weights for the similar case search but weight values are binary. The set of project attributes is variable and similarity weights depend of the knowledge search area. PM KBCS ensures list of the knowledge search area related knowledge with statistical information about frequency. In the knowledge-based logistics strategy system data warehouse and OLAP is used for CBR implementation. Similar cases are searched in the data warehouse according to defined parameters and full case description is in the repository. This KBS ensures numeric data (logistic strategy) and symbolic data (logistic workflow) processing. In the proposed PM KBCS case attributes and classification are stored only in a relational database structure, but also full case description is in the repository. PM KBCS also ensures structured and unstructured knowledge store. Also in KBS is used: ontology, a neuron network, a data mining and other artificial intelligence methods. Ontology is a shared description of the concepts and relation in domain knowledge. Ontology in KBS is used as storage system [24] and knowledge representation method [25] [26]. Data mining is a process of extracting patterns from data and is an important tool to transform data into information. Data mining techniques is used CBR case clustering/classification [18] and for knowledge mining [27].

4

Architecture of the PM Knowledge-Based Configuration System

In order to support knowledge utilization in the PMIS configuration, PM KBCS is elaborated. Principles of case-based reasoning are used in design of PM KBCS. The architecture of PM KBCS is shown in Figure 2. Cases in PM KBCS are a) empirical knowledge or configuration files Cj, j=1,…, m, that previously have been used for configuration of PMIS and b) PM methodologies, standards and best practices Hi, i=1,…, p. The CBR process is managed using the configuration client, which has three main modules. The first module is used to describe a new case using set of the project attributes Aj+1, where j+1 is used to identify the new case. The retrieve step is performed by the information retrieval module that find the similar cases according to the information search knowledge area Ms (e.g., risk management) and the defined similarities Lk, k=1,…, n. Sets of cases similar to the new case j+1 and search knowledge area s are denoted by Hj+1,s’ and Cj+1,s’ for theoretical end empirical knowledge, respectively, is the result of information retrieval. The information processing and display module performs CBR reuse and revise steps. This module collects and processes gathered information and displays it to the user. CBR retain or learning is performed after the new configuration file Cj+1 has been used for configuration on PMIS (Figure 1, transformation T2k). Cj+1 is added to the PM knowledge repository as a new case.

Figure 2. PM KBCS architecture - 33 -

4.1

PM Knowledge Repository The PM knowledge repository is conditionally divided into two parts: the library and the case register (Figure 2). In the library, knowledge is stored in three ways: theoretical (H = {Hi|i=1,…, p}), empirical (C = {Cj|j=1,…, m}) and unstructured (D = {Dl|l=1,…, s}). Unstructured knowledge is in various formats e.g. diagrams, forms templates or documents, but this information in some ways is related with data stored in H or C. The case register collects information that is needed for knowledge organization, search and retrieval. It stores information about the case descriptions (PH = {PtH| t=1,…, x} and PC = {PtC| t=1,…, y}), the project attributes Aj, its values and the case similarity (L = {Lk| k=1,…, z} ). Knowledge classification is ensured by the set of project attribute Aj= (a1, …, an). Each attribute ai also has defined values. Attribute default values, mandatory and multi-choice are stored with the attribute configuration. XCPM schema [10] defines the set of typical attributes represented using the EnvironmentFactors element (Figure 3). This element also allows storing of the additional attributes with the OtherFactor element. Each case description PtH or PtC includes a set of case describing attributes Ai and the case identifiers Hi’ or Cj’. The case description PtH=(Ai, Hi’) defines the theoretical knowledge case Hi, but PtC=(Aj, Cj’) similarly describes the empirical cases Cj. In situation when in the case description contains attributes with multi-choice values, the case has more than one case description according to a number of possible combinations of attribute values.

Figure 3. XCPM schema fragment: EnvironmentFactors

The case similarity Lk=(Mk, Bk, Xk) depends of the knowledge search area Mk and the knowledge type Bk, k= 1, …, n. The knowledge search area specifies what PM knowledge is searched, e.g., risk register, quality criteria, change request, documents, etc. The knowledge type defines what case search Lk is provided: theoretical (H) or empirical (C). Importance of each attribute in the search of similar cases is defined by the set of the attribute similarity measurement Xk=(x1, …, xn). The count of element for the similarity measurement Xk is equal with the count of the project attributes. The value of xi can be 0 or 1. If xi=1 then cases need to compare according to the ith attribute ai during the similar case search, but if xi=0 then attribute ai is ignored. 4.2

Information Retrieval The information retrieval module ensures retrieving similar case identifiers from the case register. Input data is the set of the project attributes Aj+1 and the search knowledge area Ms, s = 1, ..., v. Output data are retrieved sets of the cases identifiers Hj+1,s’ and Cj+1,s’. The information retrieval process consists of two steps: 1. Similarity clarification. Appropriate similarity is searched in the set of similarities L according to the search knowledge area Ms. The result is a subset of the similarities:
, = {  |  ∈ and  =  and  = 1, … , } (3) 2. Case search. Using similarities from the subset Lj+1,s, similar cases are searched in PH and PC. The search in each of case sets is performed separately and search attribute similarity already can different. With type Bk in Lk is defined the case set witch similarity can be used. Consequently, the case search is performed in two stages: 2.1. Search in set in the theoretical case PH. For each Lk where  ∈ , and  = . The set of case identifier Hj+1,s’={ | ∈  } is retrieved, which attribute values aiz are equal to new case attributes values aij+1 of those attributes that similarity measurement xi=1:

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∑$"% !" |!" ∈ # and # ∈  ) ∑$"% !" |!" ∈ # and # ∈  ) , 1 if / '" |'" ∈ =  & '( & ∈  ) = '" |'" ∈ & )) (4) 2.2. Search in set in the empirical case PC. For each Lk where  ∈ , and  = *. The set of case identifier Cj+1,s’={* |* ∈ + } is retrieved, which attribute values aiz are equal to new case attributes values aij+1 of those attributes that similarity measurement xi=1: ∑$"% !" |!" ∈ # and # ∈  ) ∑$"% !" |!" ∈ # and # ∈  ) , 1 if / '" |'" ∈ = + & '( & ∈  ) = '" |'" ∈ & )) (5) 4.3

Information Processing and Display The information processing and display module processes the similar cases found and provides recommendation for configuration of PMIS. For example, if a user configures the risk list, then module shows data elements used in similar projects or chosen methodologies. During information processing all similar cases from the sets of case identifier Hj+1,s’ and Cj+1,s’ are found in the library and information about the search knowledge area Ms is extracted for further processing. To ensure that the user is not burdened with too much information and important information is shown at first, following activities are carried out during information processing process: 1. Grouping of information in order to ensure that every information element is displayed only once. Information element value analysis is not undertaken only grouping by the full-text. 2. Collection of statistic about the each information element. Statistic includes frequency estimates expressed in % between similar cases. This statistic is also supplemented with information element associated methodologies/standards listing. 3. Information element ranking in descending order to ensure that statistically most frequent used information is displayed first. During describing PMIS configuration Cj+1, knowledge is shown in a form of suggestions and users either use of ignore these suggestions.

5

PM KBCS Application Example

An example is used to demonstrate the knowledge assisted configuration process. This example shows configuration of the risk register as a part of the PM environment setup. The risk register is used to name all risk factors relevant to the project, and it is implemented as a list containing multiple data elements. The knowledge assisted configuration helps to determine what data elements should define in the risk register. The risk register is prepared for the following project: IT company realized outsourcing maintenance project for government software. The project team includes 7 specialists and year budget is EUR 80000. The project is organized in an iterative manner and works according the PMBOK methodology and complies with ISO quality standards. The first step of the knowledge assisted configuration process is definition of the project attributes Az and their values (Figure 4).

Figure 4. Definition of project attributes Az

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The case similarity is clarified from the set of similarity L (Figure 5). In the example, the search knowledge area Ms is the risk register. The search result is the subset that includes two similarities: L5 and L6. These similarities further are used for case search. Similarity L5 is used for search in the set PH (B5=H), and L6 is used for search in the set PC (B6=C). Cases in the set PH (Figure 6) are searched after methodology (a1) because only the similarity measurement x1 is equal to 1 for the similarity L5 (Figure 5). Methodology ‘PMBOK’ is mentioned only in the case description P9H (H9’=1256). Cases in the set PC (Figure 7) are searched after 6 attributes: methodology, area, type, client type, project type and action. Two cases form the set PC are the same abovementioned attributes values as new project: P2C (C2’=A152) and P6C (C6’=A962). Three cases are found as the result of the information retrieving process.

Figure 5. Set of the similarity L

Figure 6. Set of the theoretical knowledge case descriptions PH

Figure 7. Set of the configuration knowledge case descriptions PC

During information processing, the risk register configuration information is extracted from the full case description (Figure 8). This configuration information includes information about the risk register data elements. Then all found risk register data elements are grouped and ordered. The result for a user is shown as in Figure 9. The user sees the list of data elements (i.e., data columns) that is ordered descending, statistic about frequency and methodology if this column is already mentioned in it. The column configuration information also can be analyzed and shown. Knowledge processing and display result (Figure 9) shows that such columns as ID, description, condition and owner are mentioned in all cases found in the knowledge repository. If the other information, e.g., change requests, is prepared for the described project than the sets of similar cases (Hj+1,s’ and Cj+1,s’) can be changed because a different knowledge search area (Mk) is considered and subsequently other similarity measurements (Xk) are used.

Figure 8. Risk register column lists of retrieved case

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Figure 9. Result knowledge shown for user

6

Conclusion and Future Work

Having improvement of efficiency of PMIS configuration and implementation as an overall goal, the architecture of knowledge-based configuration system for PMIS has been described in this paper. This architecture is a part of the PMIS configuration process and provides users with appropriate configuration and PM knowledge. The KBCS architecture consists of two parts: the configuration client that provides case-based reasoning for knowledge search and processing, and the PM knowledge repository that include the case register and the library. The project attributes and environment factors have been used for the case classification. During implementation and configuration of PMIS, CBS generates configuration suggestions on the basis of historical data and theoretical knowledge. That helps process designers and project managers to get configuration guidance. In the example explored, the user can see data elements frequently used in setting-up the risk register. Configuration of PMIS and accumulation and use of historical data are greatly aided by the relatively well-defined scope of project management. The XCPM schema is used to standardize definition of the project management domain. The paper explores only data similarity. However, processes play important part in PMIS. Using knowledge about processes and analyzing process similarity are main future research directions.

7

Acknowledgement

This work has been supported by the European Social Fund within the project „Support for the implementation of doctoral studies at Riga Technical University“.

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Bērziša S., Grabis J. An Approach for Implementation of Project Management Information Systems. Information Systems Development: Towards a Service Provision Society, Springer, 2009, 423-431.

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COMPUTER DECISION SUPPORT SYSTEM MODELLING SELF-ORGANISATION EFFECTS BASED ON THE ANALYSIS OF EXPERTS’ GOALS Igor Kirikov, Alexander Kolesnikov, Sergey Listopad Kaliningrad branch of the Institute of Informatics Problems of the Russian Academy of Sciences, 5 Gostinaya street, Kaliningrad, Kaliningrad region, Russia, [email protected], [email protected], [email protected] Abstract: The authors consider one of the approaches to creation of an intelligent computer decision support system, simulating self-organisation, based on the experts’ goals analysis. They also describe a universal structure of a multi-agent system, as a decision support system model for carrying out computing experiments for research into dependence of self-organising processes on the relationships of the agents’ goals. The results of these experiments are presented, which allow forming a fuzzy knowledge base of the decision-making agent for selecting a multi-agent system architecture according to the current problem properties. Keywords: computer decision support system, self-organisation, similarity measure of agents’ fuzzy goals, complex travelling salesman problem.

1

Introduction

The self-organisation concept has been one of the most outstanding and promising directions in the scientific life in the last few decades. The research of self-organisation is based on its multi-disciplinary nature and systemic approach. The self-organisation ideas go back to the works of I. Kant (who suggested that parts of a whole are mutually coordinated as causes defining each other), G.W.F. Hegel (who proposed the law of the Absolute Idea’s evolution, applicable to describing self-organising systems’ evolution), and – in the 20th century – A. A. Bogdanov (who proved – within tectology – the principle of similarity of systems’ structures and various systems’ evolution), L. Bertalanffy (who proposed the open-system concept), E. Shrödinger (who considered the negative entropy import from the environment as a driving force of self-organisation processes), I. Prigogine and I. Stengers (who considered self-organisation processes far from thermodynamic equilibrium), H. Haken (who founded synergetics, the science of coordinated interaction of system’s parts), A. P. Rudenko (who worked out the continual self-organisation theory), H. Whitney (who proposed the singularity theory), B. B. Mandelbrot (who elaborated the fractal geometry), R. Thom (who proposed the catastrophe theory), etc. Self-organisation processes are typical for the world we live in [1, 2]. For example, Benard cells (the structure that arises in a thin layer of viscous fluid at critical temperature difference between upper and lower layer surfaces, Figure 1, a); the laser emission (the set of non-correlated wave trains of the pumping lamp is transformed into a single laser sine-wave train, Figure 1, b), the reproduction process of the fungus myxomycete (where at the reproduction stage a set of individually living cells, gathers at some point, specializes, and forms a mushroom, which then throws out spores, and the life cycle recommences, Figure 1, c). b) Lamp

a)

Laser

c)

Figure 1. Examples of self-organisation in nature and technology: a) Benard cells; b) laser emission; c) life cycle of the fungus myxomycete

The above processes have common features of self-organising systems [1]: •

Endogenous global order. The system reaches a stable global state due to internal processes.

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•

Emergence. The functionality of the system as a whole is not limited to the sum of functions of its elements.

•

Self-maintenance is a capacity of the system to repair itself, to reproduce or repair its components.

• Adaptivity. The re-organisation capability implies adaptation to environmental variations. Self-organisation manifestations extend beyond the above phenomena. There are works on the selforganisation theory in the field of medicine, art, history, linguistics, and management theory. Speaking of selforganisation in the management systems, it should be noted that self-organisation arises in the process of collective decision-making in decision support systems (DSS). If self-organisation effect arises in a DSS, its decisions become qualitatively better than those of individual experts, and the system itself becomes more flexible, reliable, and efficient. Due to this effect the DSS is able to solve more complex problems than any single DSS’s expert or even all experts working separately. A conceptual DSS model [5] is presented in Figure 2. The arrows, connecting experts, show their versatile interaction. Some experts are subordinated to one or several other experts with regard to the job, i. e. there can be an organisational structure in a DSS. Co-operating in the process of discussions, the experts exchange data, knowledge, explanations, and partial solutions of the general problem. There can be teams of experts, not bound by subordination. Among them, there can be explicit or implicit leaders, which even more ‘escalates’ the heterogeneity of collective decision-making.

Figure 2. A conceptual DSS model

Each team member, expert or decision-maker (DM), listens to other participants and expresses his/her own opinion. Generally, each DSS participant has his/her own vision of the problem as well as a way to solve it and tries to convince the team members of his/her being right. The decision-making process in the DSS consists in searching for a compromise, controlled by the DM. The purpose of the search is to find ‘a state of resonance’ of the course of discussion in the DSS, which would lead to occurrence of a synergetic effect, when the collective integrated solution quality appears better and is free of defects of individual experts’ opinions. The occurrence of the synergetic effect, explaining the higher quality of collective decisions in comparison with those made by individual experts, depends on the DM’s efficiency in establishing one kind and breaking off other kinds of relations among the participants during the debates. Therefore, it is relevant to develop an automated system, simulating the DSS operation and the arising self-organisation effect (synergistic effect), resulting in a joint DSS’s decision qualitatively superior to any individual solutions. One of such systems is a multi-agent system (MAS) with self-organisation based on the analysis of the degree of interaction among the agents [4]. It is capable of simulating the behavior of a real DSS, allowing to impartially investigate the dependence of the self-organisation effect on the similarity degree of the agents’ goals and to make recommendations to decision-makers. Self-organisation modelling in this MAS is feasible by means of inclusion of a decision-making agent, performing, among other things, the function of 'interaction analysis' considered in [4]. This function allows attributing the MAS architecture to one of three classes, with regard to the similarity degree of the fuzzy agents’ goals: an MAS with cooperating agents, an MAS with neutral agents, and an MAS with competing agents. The decision-making agent, using its own fuzzy knowledge base, selects one of the architectures to solve the problem, depending on its parameters. To formulate the knowledge base rules, a series of experiments is to be carried out for determining the dependence of the self-organisation effect occurence probability on various parameters of the problem. In order to separately investigate the dependence of the MAS solutions' effectiveness on the class of the MAS architecture, based on the degree of interaction among the agents (excludng the influence of other factors) a multi-purpose MAS structure with the same number of agents and their operation algorithms for construction of an MAS will be used. This is considered in the next section,. Thus, the multi-agent systems with various architecture types, involved in the experiments, will differ only by the agents’ goals. The MAS was tested on the - 40 -

example of a complex travelling-salesman problem (CoTSP), described in the third section, and the results of this test are presented in the fourth section.

2

The structure of a multi-agent system for studying of the self-organisation effect

In carrying out our experiments, we shall consider three classes of MAS architectures with regard to the agents’ interaction degree: with cooperating agents, with neutral agents, and with competing agents. The creation of such an MAS will be based on a universal MAS structure for construction of hybrid intelligent systems (HIS) proposed in [7], which is shown in a specified form in Figure 3. In this figure, solid and dashed arrows are relationships between the agents, such as information inquiries, assistance in solving sub-problems, transfer of their decision results, etc. The chain lines show the agents’ interaction with the ontology.

Figure 3. A versatile MAS structure for construction of hybrid intelligent systems

Let us consider the designed goal of each agent in this structure in detail: 1. The interface agent interacts with the user through the input/output subsystem, inquires the information required for the problem solving and informs the user about the system’s work result. 2. The decision-making agent is responsible for the activation and synchronization of different agents. In accordance with its planning algorithm, the agent develops work plans for solving of the problem, received from the interface agent, and verifies whether the plans are implemented. The decision-making agent sends the data, required for solving of the problem, to the problem-solving agents and determines the order of their interaction. As soon as the latter have solved their subtasks, the decision-making agent selects one of the alternative decisions, and passes it to the interface agent, or starts the next iteration of the problem-solving by transferring the solution, obtained by each problem-solving agent, to all other problem-solving agents for improvement. In the process of the new iteration, the agents try to improve the solutions, obtained by the previous one. Thus, the decisions of the problem-solving agents are aggregated. The 'responsibility' for organising a synergetic effect in the MAS lies with the decision-making agent. Using the 'interactions analysis' function [3], it analyses the agents' goals and specifies the type of the MAS architecture (with neutral, cooperating, or competing agents), which will continue solving the problem. It selects one of the alternative decisions, delivered to it by the problem-solving agents, and subsequently transfers the selected decision to the interface agent. 3. The problem-solving agent is a specialized agent with its own knowledge base in the domain, required for solving the part of the problem, assigned to it. 4. The mediator agent tracks the names, models, and capacities of all registered intelligenttechnology agents. The MAS agents can refer to the mediator agent, to find out which of the intelligent technology agents can assist them in solving the part of the problem, assigned to them. 5. The intelligent technology agent provides 'services' to other agents with the use of some homogeneous or heterogeneous intelligent technology algorithms. The agent receives tasks from the problem-solving agents or other intelligent technology agents and sends back its work results. In addition, the agent can apply for other intelligent technology agents’ assistance. Before such agent can start working in the MAS structure, it must be registered with the mediator agent, i. e. provide information about the subtasks, which it is capable of solving, and about the intelligent technology, it uses. - 41 -

6.

The ontology is the basis of the agents’ relationships. The agents interpret the meaning of the received messages, based on this ontology. In addition, the ontology determines when a problemsolving agent or an intelligent technology agent requires other agents’ assistance. In an MAS with such structure, each problem-solving agent can apply any intelligent technology, represented by the intelligent technology agents, present in the system. The presence of a mediator agent in the versatile structure implies an adaptive MAS organisation, which allows moving, adding, deleting or substituting an intelligent technology agent (e.g., when an agent capable of performing the same task in a better way is present in the system). The presence of the mediator agent also ensures additional MAS stability. For example, if one of the intelligent technology agents cannot be used, the problem-solving agent can make use of assistance of another agent with the same or similar capabilities, finding it through the mediator agent. Thus, the interactions between the agents are determined directly during the MAS operation.

3

Description of the test problem

The use of this MAS structure allows creating an MAS for solving problems, which are complex in modelling and whose solution requires application of several intelligent technologies. An example of such problems is a complex travelling salesman problem (CoTSP) [6]. The CoTSP is based on the classic travelling salesman problem (ClTSP) but it takes into account many real-world factors, which are left 'overboard' by ClTSP. So, while in a ClTSP it is required to find the least-expensive route for a travelling salesman to visit N clients (cities), in a CoTSP it is required to find a set of routes for several vehicles, to be optimal by four criteria: the total cost of the set of routes; the total travel duration for all vehicles; the probability of a tardy arrival to at least one customer; and the reliability. The CoTSP takes into account such stochastic factors as the probability of occurrence of traffic jams resulting in the probability of a tardy arrival to a customer, the probability of losses due to cargo breakage, etc. The ClTSP and CoTSP graphs are presented in Figure 4 for comparison. The differences between the graphs are shown in bold. A complex travelling salesman problem is approximation of the ClTSP to the practical problem of delivering goods to the customers by several vehicles. The solution of the problem covers the full range of variables and relationships, operated by the experts to establish cargo delivery routes in practice. A special method should be used to handle a variable of each type. It is also necessary to model the relationships between the variables of different types, which, according to [5], attribute the problem to a heterogeneous class of complex modelling problems. This means that the problem cannot be solved the by any one single known method.

Figure 4. The graphs of a classic travelling salesman problem (LH) and a complex travelling salesman problem (RH)

Within the synergistic approach to artificial intelligence, a system is to be built that combines intelligent technologies for processing variables of different classes. Such a system is to be able to independently design the method to solve the problem, fed to its input. For the experimental purpose, a laboratory prototype of the automated system “A multi-agent system for solving complex travelling salesman problems (MAS SZK)” has - 42 -

been developed. It allows evaluating the conditions for emergence of a synergetic effect, in particular, the generation of the systems' self-organisation emergence capability, depending on the MAS architecture class. This makes it possible to formulate the 'conditions – architecture' rules of the decision-making agent’s knowledge base.

4

The results of the experiments

The experiments with the MAS SZK were carried out using five CoTSPs with 10, 15, 20, 25, 30 cities and three MAS architectures differing by the degree of the agents' interaction: an MAS with cooperating agents, an MAS with neutral agents, and an MAS with competing agents. One hundred computing experiments were carried out with the test problems of each MAS architecture class. The parameters examined in the experiments were the percentage of the MAS solutions superior to any of the agents’ individual solutions (Figure 5) and the average value of the decision-making agent’s goal (Figure 6). The first parameter characterizes the probability of synergetic effect occurrence in the MAS. The higher is its value, the higher is the MAS efficiency. The second parameter is the average value of the function of belonging with the decision-making agent’s fuzzy goal in 100 experiments – the optimality criterion of the decisions made by the MAS. The value of this parameter varies within the interval [0, 1] – the higher is the value, the better are the decisions made by the MAS.

Figure 5. Percentage of the MAS solutions superior to any of the agents’ individual solutions

Figure 6. The average value of the decision-making agent’s fuzzy goal

As can be seen from Figure 5, regardless of the problem dimension, a synergetic effect most often occurs in the MAS with neutral agents, and in the second place – in the MAS with cooperating agents. In addition, there is a tendency to decreasing of the synergetic effect occurrence probability with the increase of the - 43 -

problem dimension. It is also noticeable that with the growth of the problem dimension, the gap in the probability of the synergetic effect occurrence between the MAS with neutral agents and other MAS architectures increases, which indicates the efficiency of interaction between agents, implemented in the MAS with neutral agents, and active influence of this interaction on the quality of decisions. Figure 6 also shows that the smaller is the dimension of the CoTSP, the lesser is the influence of the synergetic effect on the quality of decisions. Therefore, when the problem dimension is small, the MAS architecture can be chosen at random or based on the analysis of other CoTSP parameters (completeness of the adjacency matrix, topological features, etc.), while for a problem involving over 30 towns, where the arising synergetic effect begins to play a major role, the MAS with neutral or cooperating agents should be chosen. Depending on the problem dimension, these MAS architecture selection rules form a fuzzy knowledge base of the decision-making agent. This knowledge base can be extended after a study of the dependence of the synergistic effect probability on other CoTSP parameters, such as the number of roads (road network density), the topological features, the number of salesmen, etc. Besides, it is advisable to use a backward error propagation algorithm, actively used for training neural networks, to adjust membership functions of fuzzy variables of the fuzzy knowledge base during the system operation. This will allow reducing the impact of possible expert’s errors in designing of the knowledge base, as the knowledge base will be able to 'learn on its own', while dealing with practical problems in its operation.

5

Conclusion

Thus, we have explored the self-organisation effect in decision support systems. A versatile MAS structure, based on which multi-agent systems with various types of architectures for carrying out computing experiments have been implemented, is proposed. The results of the experiments with different types of the MAS architectures have been presented. In addition, we have formulated the heuristic dependence of probability of the synergetic effect occurrence on the MAS architecture and the problem dimension. Similarly, knowledge about the MAS architectures' effectiveness depending on other parameters of the problem (number of roads, road network density, topological features, the number of salesmen, etc.) can be obtained in order to form the 'condition - architecture' knowledge base. This is a subject for further research.

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Di Marzo Serugendo G., Gleizes M.-P., Karageorgos A. Self-organisation in multi-agent systems. The Knowledge Engineering Review. 2005, volume 20, №2, p. 165 — 189. Haken H. Information and Self-organisation. A macroscopic approach to complex systems. Berlin, Springer. 2006. Kirikov I.A., Kolesnikov A.V., Listopad S.V. Modelirovanie samoorganizatsii grupp intellektualnyh agentov v zavisimosti ot stepeni soglasovannosti ih vzaimodeistviya. Informatika i yeyo primeneniye. 2009, volume 3, issue 4, p. 78 — 88. (in Russian) Kolesnikov A., Listopad S., Kirikov I. Investigation of self-organisation phenomena and processes in decision support systems according to relationships of participants’ goals. “Proceedings of the 16th International Conference on Information and Software Technologies, IT 2010”. Kaunas (Lithuania): Kaunas University of Technology. 2010, p. 88 — 94. Kolesnikov A.V. Gibridnye intellektual'nye sistemy. Teorija i tehnologija razrabotki. Sankt-Peterburg, Izd-vo SPbGTU. 2001. (in Russian) Listopad S.V. Reshenie slozhnoj prakticheskoj zadachi kommivojazhera metodami gibridnyh intellektual'nyh system. “Iskusstvennyj intellekt: filosofija, metodologija, innovacii: Materialy vtoroj mezhdunarodnoj molodezhnoj konferencii. Sankt-Peterburg, 15-17 nojabrja 2007 g.”, Sankt-Peterburg. 2007, p. 199 — 201. (in Russian) Zhang Z., Zhang Ch. Agent-Based Hybrid Intelligent Systems: An Agent-Based Framework for Complex Problem Solving. LNAI 2938, Springer. 2004.

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COMBINING 2-OPT, 3-OPT AND 4-OPT WITH K-SWAP-KICK PERTURBATIONS FOR THE TRAVELING SALESMAN PROBLEM Andrius Blazinskas, Alfonsas Misevicius Kaunas University of Technology, Department of Multimedia Engineering, Studentu St. 50−401, 416a, Kaunas, Lithuania, [email protected], [email protected] Abstract. The Traveling Salesman Problem (TSP) is a famous NP-hard problem typically solved using various heuristics. One of popular heuristics class is k-opt local search. Though these heuristics are quite simple, combined with other techniques in an iterated local search (ILS) framework they show promising results. In this paper, we propose to combine the 2-opt, 3-opt and 4-opt local search algorithms with so called k-swap-kick perturbations, which are a generalization of the well known double-bridge (random 4opt) move. To reduce the run time of our 2-opt, 3-opt and 4-opt implementations, we apply some enhancements like a candidate list (based on k-d tree), search cuts, greedy starts, two-level tree data structure and others. We provide experimental results of the implemented algorithms with the TSPLIB instances. Keywords: traveling salesman problem, 2-opt, 3-opt, 4-opt, k-swap-kick.

1

Introduction

The traveling salesman problem (TSP) [6] can be formulated as follows. Given a set of cities and distances between them the task is to find shortest tour (Hamiltonian cycle) visiting every city only once. The problem where distance between two cities does not depend on the direction is called symmetric TSP (STSP) and asymmetric TSP (ATSP) otherwise. TSP is considered to be NP-hard problem. The total number of possible tours for STSP is equal to (n−1)!/2 (where n is the problem size – the number of cities (nodes)) [7]. It is a relatively complicated task to find good tours in an acceptable time for such a search space, especially when n is large. There are two main ways of solving the TSP: exact methods and heuristics. The exact methods are too time-consuming for larger n, thus heuristics typically are used. One of the leading heuristics for the STSP is LinKernighan [9]. Effective implementations of it exist (see, for example, Concorde* [2], Helsgaun's code** (LKH) [7]). In this paper, we consider only STSP. In the experiments, we use k-opt heuristics, which are closely related to a more robust Lin-Kernighan heuristic, since both use k-opt sub-moves [8]. In Section 2, we describe several ways how to optimize 2-opt, 3-opt and 4-opt heuristics to make them run faster. In Section 3, k-swap-kick perturbations are described and in Section 4 experimental results for combining 2-opt, 3-opt and 4-opt heuristics with k-swap-kick perturbations are presented. We also provide conclusions and future research ideas in Section 5.

2

Improving 2-opt, 3-opt and 4-opt speed

While 2-opt algorithm is simple and its naive form involves repeated breaking of two edges and reconnecting them in other (cost decreasing) way (see Figure 3b) until no positive gain 2-opt move can be made, when it comes to practice, there is more specifics to consider and different authors implement it differentially. For example, some authors check all possible flips and perform only the best one [11], while others simply make the first found positive gain flip (see DIMACS Implementation Challenge for STSP*** on variations). Several other possible implementation choices for 2-opt heuristic are also indicated in [3]. Such choices typically significantly impact the cost and timings. Most of these considerations also apply for 3-opt and 4-opt, but even more cases and specifics needs to be evaluated. The time complexity for naive 2-opt, 3-opt and 4-opt is O(n2), O(n3) and O(n4) respectively, however this can be greatly improved by using various speedup techniques. These techniques typically slightly sacrifice true local optimality (resulting tours are not really k-optimal), but in some cases combination of them allows reaching nearly O(n) complexity [1]. We use several important improvements for our fast 2-opt, 3-opt and 4-opt modifications (2-opt-f, 3opt-f and 4-opt-f). We are outlining them shortly in the subsequent subsections.

*

Concorde TSP solver, http://www.tsp.gatech.edu/concorde/ Helsgaun's Lin-Kernighan implementation, http://www.akira.ruc.dk/~keld/research/LKH/ *** DIMACS Implementation Challenge for STSP, http://www2.research.att.com/~dsj/chtsp/

**

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2.1

K-d tree based candidate list Nearest node candidate lists (CL) are widely used for improving k-opt heuristics [1]. K-d tree usage for CL identification is much more efficient than naive CL approach. In particular, naive CL can be generated in O(n2logn) time [11], while k-d tree − in O(Kn + nlogn) [4] (where K is the number of dimensions and in our case always K = 2). Once generated, naive CL does not require any additional processing to extract lists (it simply contains them in arrays), while k-d tree needs to be searched every time a list is needed. This, of course, can be solved by using caching, but for large TSP instances caching requires a lot of memory which grows in O(mn) (where m – CL size) and for storing k-d tree this number is roughly O(n) (efficient Concorde k-d tree implementation uses 52n bytes of memory). Searching in k-d tree (bottom-up technique) can be done in almost O(1) time [4]. Thus efficient k-d tree implementation beats naive CL practically in all cases. Benefits of using kd tree for CL are obvious, for example, on our test machine (see Section 4 for specifications) for pla85900 TSP problem instance construction of CL in naive way takes roughly 15 minutes, while construction of k-d tree with searching and caching of CL – about 3 seconds (CL size in both cases is equal to 80). It must be noted, that usage of k-d tree limits TSP solver to Euclidean instances. 2.2

Stop search if b1 = a2 Instead of a popular don't look bits idea [1], we propose using search cut if a1 nearest candidate b1 is already the next node to a1 (that is, a2 = b1) in the current tour (see Figure 2 for pseudo code). A similar cut is used for 2-opt fixed-radius search in [3]. We also use such cuts for 3-opt and 4-opt when b2 = c1 and c2 = d1. Our experiments indicate that these improvements greatly shorten running times without significantly increasing cost. 2.3

For 3-opt and 4-opt consider cases where only 3 and 4 edges are exchanged respectively While breaking k edges in a tour, there are (k−1)!2 k−1 ways to reconnect it (including the initial tour) to form a valid tour [6], typically not all of these cases are considered [15]. In our implementation, we consider only those cases where all k edges are new. For example, breaking 3 edges in 3-opt there are in total 8 cases for reconnection (Figure 1), but only 4 of them (e, f, g, h) actually introduce all new edges (all others, except initial one, are simple 2-opt moves). Similarly for 4-opt, breaking 4 edges we have 48 ways to reconnect, but only 25 cases offer all 4 new edges. This does not affect 2-opt heuristic, since there is only one way to reconnect to a valid tour.

Figure 1. All possible 3-opt reconnection cases

Cascading 3-opt and 4-opt We have noticed that cascading k-opt heuristics reduces overall running time. We exploit this feature in this way: for 3-opt-f, we additionally preprocess tours with 2-opt-f and for 4-opt-f preprocessing is done with 2opt-f and 3-opt-f in this order. Optimization is started from random node every time.

2.4

2.5

Greedy multi-fragment initial tours Multi-fragment heuristic (simply called Greedy heuristic) is an effective tour construction heuristic proposed in [3]. It is known to be particularly good as a preprocessor for k-opt heuristics and more effective than traditional nearest-neighbour heuristic [1, 3]. Using such greedy starting tours in k-opt, improvement is observed for both – tour quality and running times. In our implementation, multi-fragment tour construction is backed with earlier described k-d tree for efficiency reasons. Complexity for this heuristic is O(nlogn) [3]. It should be noted, that it makes sense to use this heuristic only once for particular problem, since there is no different starting points like with the nearest-neighbor and thus the resulting tour is always the same. 2.6

Two-level tree data structure Two-level tree [5] is one of the most popular data structures for tour representation in the TSP [1, 7]. It offers O(√) complexity for reversing (flipping) the sub-tour – one of the most common operations in the TSP. It is useful for problems where n ≤ 106, otherwise Splay trees seems to be a better choice [5]. It should be noted, that usage of two-level tree is helpful only within the optimized k-opt heuristics, where time required for flip operation becomes dominant. In our experiments for naive 2-opt, only changing array with two-level tree resulted in higher running times: traversing through nodes in two-level tree is more costly, since it requires accessing parent nodes and performing additional checks (though still O(1), the constant is actually larger), while

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for simple array representation it is only a matter of increasing or decreasing the array index. Considering above mentioned improvements, a rough pseudo code for the fast 4-opt implementation is provided in Figure 2. procedure fast4Opt(tour, cl, costs, startNode) begin localyOptimal := FALSE; while (not localyOptimal) begin localyOptimal := TRUE; a1 := startNode; repeat a2 := next(a1); for (∀b1 ϵ cl(a1)) begin if (a2 == b1) break; // speedup b2 := next(b1); for (∀c1 ϵ cl(b1)) begin if (b2 == c1) break; // speedup c2 := next(c1); for (∀d1 ϵ cl(c1)) begin if (c2 == d1) break; // speedup d2 := next(d1); findBestReconnection(a1, a2, b1, b2, c1, c2, d1, d2, costs); if (better reconnection exists) begin reconnect(a1, a2, b1, b2, c1, c2, d1, d2, tour); localyOptimal := false; end end end end a1 := next(a1); until (a1 == startNode); end end fast4Opt Figure 2. General pseudo code for fast 4-opt implementation

3

K-swap-kick perturbation

The simplest non-sequential k-opt move is a double-bridge move (4-opt move) [8] first mentioned in [9] and later recognized as an effective mutation operator [10]. It is well known move, random version of it is used in one of the most powerful heuristics – Iterated Lin-Kernighan [1]. A generalization of this move is a k-swapkick (a concatenation of k segments: s1, sk, sk−1,...,s2) introduced in [14] as an improvement for LKH. Because LKH considered improving k-opt moves of size k ≤ 5, it was proposed to use k-swap-kicks of size k ≥ 6. Since we will be applying k-swap-kicks to our fast 2-opt, 3-opt and 4-opt heuristics, we will omit this restriction. We will consider all same pattern having moves with k ≥ 3 and simple 2-opt move (k = 2). Graphical 2-opt move and k-swap-kicks illustration for k = 3, 4 is provided in Figure 3.

Figure 3. 2-opt move and k-swap-kick perturbations: a) initial tour, b) 2-opt move, c) 3-opt move (k=3) and d) doublebridge or 4-opt move (k = 4)

All k-swap-kicks are special cases of k-opt moves. Clearly for k-swap-kick to be performed, a condition of n ≥ 2k is enough. We will be analyzing only random 2-opt and k-swap-kick moves, the ones which do not consider the effect on tour cost during perturbation and may typically lead to worse tours. This is opposite to the moves used in k-opt heuristics, which typically perform only tour-cost-decreasing moves. Random double-bridge move previously was also combined with 3-opt heuristic, forming Iterated 3-opt [1]. In some cases, it even yielded better results to that of Lin-Kernighan and was slightly worse than Iterated Lin-Kernighan. We continue by extending this idea. We perform an experiment combining our 2-opt-f, 3-opt-f and

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4-opt-f implementations with random 2-opt move and random k-swap-kicks (k = 3...15). Pseudo code on how this combination is implemented is provided in Figure 4b. z* := ∞; repeat Generate a greedy randomized solution t; Find local optimum tl with local search starting from t; if (z(tl) < z*) begin z* := z(tl); t* := tl; end until a stopping criterion is satisfied;

Generate a greedy solution t; Find local optimum tl with local search starting from t; z* := z(tl); repeat Perturb tl tour using random k-swap-kick forming tlp; Find local optimum tlpl with local search starting from tlp; if (z(tlpl) < z*) begin z* := z(tlpl); t* := tl := tlpl; end until a stopping criterion is satisfied;

(a)

(b)

Figure 4. General pseudo codes for GRASP (a) and k-swap-kick perturbation (b) frameworks (t* – best solution found and z* = z(t*), where z – objective function)

It should be noted that k-swap-kick ILS framework is quite similar to other well known procedure – GRASP (see Figure 4a, also see [12]). The main distinction between them is that GRASP starts from a totally different initial tour in every iteration, rather than perturbing existing one.

4

Experimental results

All algorithms are fully implemented using Java (JDK 6). Implementations of two-level tree, k-d tree and multi-fragment heuristic are based on Concorde C code. Experiments were performed on the machine with Intel Core i7-860 processor and 8GB of RAM. Below provided tables and graphs summarize the results of our fast 2-opt, 3-opt and 4-opt implementations (2-opt-f, 3-opt-f and 4-opt-f). To emphasize the advantage of perturbations, we also performed two additional experiments without using perturbations: one with random starts and other with greedy starts. In these experiments, we used similar to GRASP procedure, but it did not fully comply with basic GRASP scheme (see Figure 4a): random start experiment did not use greedy randomized heuristic for initial solution and greedy start experiment had no randomization (since we were using original multi-fragment heuristic). The only randomization in later case was different start point (node) in fast k-opt local search. All other consecutive columns provide the results for random 2-opt (k = 2) and k-swap-kick perturbations with fixed k. In both frameworks, the number of iterations was set to 1000 (stopping criterion). Deviations from the optimal tour length (optimal tour lengths can be found in TSPLIB [13]) were estimated using formula: δ = 100( z * − zopt ) zopt [%] , where z* – obtained best value of the objective function and zopt denotes the provably optimal objective function value. All experiments were repeated for 10 times and averages calculated. The problem data for experiments were taken from the TSP library – TSPLIB [13]. Table 1. Experimental results for 4-opt-f: tour quality (average deviation from optimal of 10 runs, %) Problem name eil51 st70 kroE100 kroB150 ts225 gil262 a280 lin318 rd400 u574 rat783 vm1084 pcb1173 vm1748 d2103 fnl4461 rl5934

4-opt-f Random start 0.00 0.00 0.03 0.19 0.21 1.73 1.81 1.66 2.42 2.98 3.96 3.05 4.92 3.68 6.33 4.96 5.71

Greedy start 1.17 3.26 0.00 2.11 1.35 2.11 1.98 2.09 2.10 2.50 2.58 1.92 3.78 2.89 2.46 2.57 2.95

4-opt-f with greedy start and k-swap-kicks k 2 3 4 5 6 10 0.02 0.07 0.05 0.02 0.12 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.02 0.00 0.00 0.11 0.02 0.04 0.04 0.02 0.03 0.02 0.00 0.00 0.00 0.00 0.00 0.40 0.46 0.13 0.14 0.23 0.20 0.64 0.48 0.02 0.05 0.20 0.03 0.54 0.49 0.36 0.42 0.35 0.37 0.57 0.33 0.24 0.33 0.29 0.51 1.18 0.53 0.62 0.63 0.69 0.79 1.43 1.00 1.11 1.13 1.19 1.55 0.87 0.47 0.47 0.60 0.50 0.77 2.10 1.76 1.45 1.77 1.82 2.15 1.24 0.81 0.85 0.96 1.08 1.23 1.20 0.91 0.82 1.22 1.22 1.53 2.21 2.19 2.17 2.44 2.47 2.76 2.06 1.82 1.81 2.21 2.08 2.69 - 48 -

15 0.02 0.00 0.02 0.03 0.00 0.27 0.04 0.54 0.54 1.06 1.78 0.98 2.54 1.52 1.80 2.74 3.07

Best found (k) 0 (all) 0 (all) 0 (all) 0 (all) 0 (all) 0 (4, 6) 0 (all) 0 (2, 4) 0.07 (4, 6) 0.13 (3) 0.48 (3) 0.15 (3, 4) 1.17 (4) 0.50 (4) 0.27 (4) 1.92 (2) 1.47 (3)

Problem name pla7397 rl11849 usa13509 brd14051 d15112 d18512

4-opt-f Random start 4.85 6.30 5.11 5.30 5.25 5.37

Greedy start 2.94 2.85 3.07 3.03 2.92 2.84

4-opt-f with greedy start and k-swap-kicks k 2 3 4 5 6 10 2.01 1.74 1.62 2.04 1.86 2.28 2.52 2.58 2.52 2.82 2.88 3.14 2.61 2.72 2.59 2.98 2.93 3.21 2.77 2.89 2.94 3.07 3.09 3.06 2.71 2.82 2.87 2.94 3.01 3.10 2.68 2.82 2.84 2.94 2.94 3.04

15 2.71 3.13 3.19 3.27 3.16 3.07

Best found (k) 1.34 (4) 2.23 (4) 2.47 (2) 2.66 (2) 2.58 (2) 2.54 (2)

Figure 6. Running time dependence on k-swap-kick size (problem d18512)

Figure 5. Running time dependence on problem size (k-swap-kick size = 4, logarithmic scale)

Table 2. Experimental results for 4-opt-f: running time (average of 10 runs, seconds) Problem name eil51 st70 kroE100 kroB150 ts225 gil262 a280 lin318 rd400 u574 rat783 vm1084 pcb1173 vm1748 d2103 fnl4461 rl5934 pla7397 rl11849 usa13509 brd14051 d15112 d18512

5

4-opt-f Random start 0.51 1.04 1.98 2.79 3.09 5.74 5.15 10.46 9.52 17.71 18.21 31.18 34.78 61.29 65.52 175.72 238.15 964.15 716.1 1197.66 894.35 1018.65 1159.27

Greedy start 0.33 0.36 0.86 1.53 0.73 1.73 1.73 4.52 3.37 9.28 6.32 16.08 13.31 33.66 16.4 59.5 88.53 377.13 241.73 619.04 347.39 474.92 510.4

4-opt-f with greedy start and k-swap-kicks k 2 3 4 5 6 0.07 0.16 0.17 0.2 0.21 0.19 0.3 0.36 0.35 0.41 0.27 0.46 0.58 0.6 0.76 0.37 0.58 0.69 0.71 0.87 0.32 0.56 0.56 0.66 0.78 0.57 1.01 1.06 1.18 1.47 0.37 0.65 0.67 0.73 0.98 1.3 2.26 2.45 2.71 3.17 0.87 1.65 1.63 1.89 2.28 1.74 3.12 3.19 3.55 4.35 1.31 2.49 2.61 2.97 3.59 2.76 5.26 5.2 5.98 7.07 2.38 4.5 4.31 5.32 6.39 5.51 10.04 9.65 11.34 13.78 3.58 6.35 6.29 7.24 8.69 11.44 22.77 22.74 26.38 32.31 16.11 31.66 29.63 37.66 42.11 50.73 118.11 111.6 143.28 170.82 39.93 86.49 80.5 101.43 119.85 89.98 186.62 195.33 222.92 269.8 62.32 132.13 130.7 143.25 176.97 70.16 144.26 144.54 163.4 195.59 77.3 156.8 160.93 193.64 215.89

10 0.29 0.53 0.93 1.08 1.05 1.8 1.33 4.63 3.35 6.59 5.46 10.72 9.8 20.37 12.12 38.78 56.06 222.82 149.98 318.19 204.42 233.7 253.28

15 0.32 0.67 1.26 1.45 1.39 2.54 1.82 5.58 4.07 7.74 6.56 12.57 12.19 21.04 14.63 44.42 67.38 278.1 172.37 356.73 228.78 264.27 265.59

Conclusions and future research

In this paper, we have proposed efficient implementations of 2-opt, 3-opt and 4-opt heuristics for solving the traveling salesman problem. We were also concerned with combining these heuristics with the random 2-opt move and k-swap-kick perturbations. We provide the results of the experiments with these heuristic variants as well. To show efficiency of the perturbations-based approach, a simplified GRASP-like procedure was, in addition, used in the comparison of the heuristics. - 49 -

As can be seen from the experimental results, the bigger problem size is − the less effect (in cost terms) perturbations have and simply increasing the perturbation size only increases cost and running time. Overall perturbation algorithm processing time dependence on problem size is approximately O(n1.2), while dependence on kick size − logarithmic (see Figure 5 and Figure 6). Because of absence of randomness in multi-fragment heuristic, such greedy starts for k-opt heuristics, without using perturbations, are not so effective for smaller problem sizes. This can be seen on the instances with n < 400, where simple non-improved random starts give much better results, however going beyond that limit, greedy starts surpass. Though, as expected, the k-swap-kick of size 4 (i.e. double-bridge move) proven to be one of the most effective perturbations, overall best solutions were also often found by using simple random 2-opt move and 3opt move perturbations (see Table 1, last column). This proposes idea for further experiments, where doublebridge moves could be combined with 2-opt and 3-opt moves in some reasonable manner. Another interesting extension would be usage of quadrant nearest neighbor CL [6] instead of typical CL. However, this may not be effective with the cut optimization described in this paper (Section 2.2), so further improved approaches may be needed.

References [1] Aarts E., Lenstra J. K. Local search in combinatorial optimization. John Wiley & Sons, Inc, 1997. [2] Applegate D. L., Bixby R. E., Chvátal V., Cook W. J. The traveling salesman problem: A computational study, Princeton University Press, 2007. [3] Bentley J. L. Fast Algorithms for Geometric Traveling Salesman Problems, ORSA J. Comput., 1992, vol. 4-4, 387-411. [4] Bentley J. L. K-d trees for Semidynamic Point Sets. Proceedings of the 6th Annual ACM Symposium on Computational Geometry, 1990, 187-197. [5] Fredman M. L., Johnson D. S., Mcgeoch L. A., Ostheimer G. Data Structures for Traveling Salesmen. Journal of Algorithms, 1995, vol. 18-3, 432-479. [6] Gutin G., Punnen A. P. The Traveling Salesman Problem and Its Variations, Kluwer, 2002. [7] Helsgaun K. An effective implementation of the Lin–Kernighan traveling salesman heuristic. European Journal of Operational Research, 2000, vol.126, 106-130. [8] Helsgaun K. General k-opt submoves for the Lin-Kernighan TSP heuristic. Mathematical Programming Computation, 2009, 119-163. [9] Lin S., Kernighan B. W. An effective heuristic algorithm for the traveling-salesman problem. Operations Research, 1973, vol. 21, 498-516. [10] Martin O., Otto S. W., Felten E. W. Large-Step Markov Chains for the Traveling Salesman Problem. Complex Systems, 1991. [11] Misevičius A., Ostreika A., Šimaitis A., Žilevičius V. Improving Local Search for the Traveling Salesman Problem. Information Technology And Control, Kaunas, Technologija, 2007, Vol. 36, No. 2, 187 – 195. [12] Pitsoulis L. S. and Resende M. G. C. Greedy randomized adaptive search procedures. In P.M. Pardalos and M.G.C. Resende, editors, Handbook of Applied Optimization, Oxford University Press, 2002, 178-183. [13] Reinelt G. TSPLIB — A traveling salesman problem library. ORSA Journal on Computing, 1991, vol.3-4, 376-385. Access via the Internet:: .] [14] Richter D., Goldengorin B., Jager G., Molitor P. Improving the Efficiency of Helsgaun's Lin-Kernighan Heuristic for the Symmetric TSP. In Proceedings of the 4th conference on Combinatorial and algorithmic aspects of networking, 2007. [15] Sierksma G. Hamiltonicity and the 3-Opt procedure for the traveling salesman problem. Applicationes Mathematicae, 1994, vol. 22-3, 351–358.

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NEW DECISION MAKER MODEL FOR MULTIOBJECTIVE OPTIMIZATION INTERACTIVE METHODS Andrejs Zujevs1, Janis Eiduks2 1

Latvia University of Agriculture, Department of Computer Systems, Liela street 2, Jelgava, Latvia, [email protected] 2 Riga Technical University, Department of System Theory and Design, Meza street 1/4, Riga, Latvia, [email protected] Abstract. Decision Maker Model instead of human Decision Maker can be used for testing/comparing multiobjective optimization interactive methods. New Decision Maker Model (called ZuMo) was defined as two criteria multiobjective optimization problem which is solved by using multiobjective evolutionary algorithm NSGA–II. Model can significantly reduce time and workload of testing/comparing experiment design and implementation and easy can be used for ad–hoc methods. Evolutionary algorithms are effective in parallel computing that can reduce experiment time also. The designed model was tested comparing STEM and GUESS interactive methods solving three 2D criteria and three 3D criteria testing problems. Different metrics are used: iteration count, stopping reason, general distance, error ratio, spacing and maximal Pareto error. The GUESS method was more effective in obtaining goal solution then STEM method. The new ZuMo model is universal and can be updated for testing/comparing different multiobjective optimization interactive methods and ad–hoc methods also. Model framework if necessary can be extended by defining additional criteria. Keywords: Decision Maker model, multiobjective optimization, comparative study, interactive method.

1

Introduction

Multiobjective optimization (MO) solves problems with two or more conflicting criteria functions. The MO methods are classified: a priory, a posteriori and interactive. Solving optimization problem by the interactive multiobjective optimization method Decision Maker (DM) provides preference information for optimization method before iteration. If the DM is satisfied with the obtained solution optimization stops, otherwise the DM sets the new preference information and method solves problem once again. There are many studies where the interactive optimization methods are compared with group of DM, usually more than 65 peoples. Experiment design and implementing is time consuming, especially for testing/comparing ad–hoc optimization methods. Other reason is DM ability to understand problem nature if it solved more times. Decision Maker Model (DMM) can prevent such problems. The MO problem (MOP) definition is [1]: (1) minimize f x , f x , … , f x  g  x  0; h x  0; x  S  where objective functions f : R R. We denote the objective vector f x  f x , f x , … , f x . The variable vectors x  x , x , … , x  belongs to the feasible region (set) S formed by inequality and equality constraints g  x and h x . Interactive optimization method (IOM) is a part of interactive optimization procedure (Figure 1.). Optimization starts with initial solution x ! and DM at each iteration defines preference information. Further optimization problem can be updated accordingly to the preference information and method solves (1) optimization problem. If the DM is satisfied with solution, then optimization stops. Otherwise, DM sets i  i " 1 and provides the new preference information to continue optimization process. It should be noted that DM define the new preference information by using values of the previously obtained solutions or graphical information (diagrams). Structure and type of preference information are depended on optimization method. GUESS method has reference point as preference information, while STEM uses grouping of criteria in two classes – improvable and relaxed. The base of reliable comparison of IOM is large amount of IOM test results. IOM testing with human DM has two main difficulties: first, necessity of large amount of DM and high workload of experiment design and implementation. 98 DM for CONTEX method testing [2]; 36 DM for GDF, STEM and Trial–And–Error methods comparison[3]; 24 DM for SIMOLP, GUESS and Tchebycheff methods comparison in [4]; 84 DM for ZW, GUESS and SMART methods comparison [5]. Second, after solving a problem several times DM acquires information about MOP properties that reduces reliability of experimental data [6]. As alternative to human DM some works suggest usage of linear and nonlinear value function instead of DM. In [7] work was used nonlinear value function for GDF, STEM and SWT methods comparison; in [8] work was used linear and nonlinear value function for Tchebysheff and SIMOLP method comparison; in paper [9] is used value function together with human DM. Unfortunately a value function no universal for different IOM and - 51 -

rise many problems in data comparison. Value function has strong decision making behavior then human DM can be sometimes inconsequent. START

Initial solution + ! i=1

charts, i=i+1 DM

STOP

Optimization problem Optimization method

numerical

Preference

Charts and data storage

* + , , + ,

Figure 1. Interactive optimization procedure

The GUESS and STEM methods was used because they has different preference information types. STEM (STEP method) firstly introduced in [2] and GUESS firstly introduced in [10].Using STEM method DM specify nadir and ideal criteria vectors and divides objectives into two classes $ % – objectives functions to improve and $ & – objectives function to relax with upper bounds for each function. Using GUESS method DM specify nadir and ideal criteria vector, upper and lower bound for all criteria and reference point with values between values of nadir and ideal criteria vector. Depending on MO method testing type – human and mathematical metric differs. In experiments with human DM use: 1) DM satisfaction of last obtained solution; 2) easy of 'handling'; 3) simple to understanding; 4) support of decision making; 5) iteration count; 6) spent time; 7) preference information impact of obtained solutions[11]. In experiments with DM model (DMM) use: 1) iteration count; 2) spent time; 3) Euclidian distance to the goal solution; 4) general distance to the theoretic Pareto front; 5) spacing [12]. Mathematical approach reveals mathematical characteristics of methods, but not gives information about method usefulness or easy to understanding for DM. The aim of the study is to create a DMM that is universal for different IOM and gives possibilities of understanding IOM methods from mathematical and DM sides. The model is validated by GUESS and STEM methods solving typical optimization problems and six different metrics are used for comparison. The DMM was defined as multiobjective optimization problem, which was solved by a variant of NSGA–II algorithm reported in [13].

2

Decision Maker Model

As mentioned before, human DM obtains experience about optimization problem, if it will be solved a lot of times. This fact negatively impact analyses of experimental data. Additionally, in experiments a large amount of the DM, usually more than 65 people, are necessary. IOM comparison with DMM gives many advantages: 1. not necessary to use human DM; 2. time of experiment design and implementation can be decreased; 3. possible to compare ad–hoc optimization methods; 4. different mathematical metrics can be used; 5. can be easy to compare new optimization methods; 6. can be easy to implement experiments. DMM (called ZuMo) was defined as two criteria MOP which will be solved by using one of multiobjective evolutionary algorithms. First criterion evaluates Euclidean distance from the current solution f'x  ( to the goal solution z )  f'x ) (. This criterion force DMM to obtain explicitly defined goal z ) . The second criterion evaluates the preference information correctness accordingly to the optimization method requirements.

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ZuMo definition is:





min-p 'x  , z ) , VRI1 (, p 'VRI1 (2

(2)

,

x  : R  S;z ) : R  D;45$6  789:



where + , – solution of testing problem (TP) for i iteration, z ) – goal solution of TP, VRI1 – preference information for i iteration and problem (2) variables; S – feasible region (set) of TP variables, D – feasible region (set) of TP criteria functions, T – testing problem, S; – correctness of VRI1 . First criterion can be defined:





p 'x  , z ) , VRI1 (  ?∑AD 'zA) B uA ( , x  : R  S; u, z ) : R  D

(3)



where x  – solution of testing problem (TP) for i iteration, z ) – goal solution of TP, VRI1 – preference information for i iteration and variables of (2) problem, k – number of TP functions, n – number of TP variables, uA – value of j criterion of TP. The values of uA can be obtained by solving optimization problem:



u  F1 'f'x  (, VRI1 (,

+ : F  7, *'+ (: F  I ,

,

G

,

(4)

H

 VRI1  S; value for the STEM method was evaluated by checking these cases: 1) add 1 if TP criteria index presents in $ % (objectives to improve) or more times (duplicates) in $ & (objectives to relax); 2) add 2 if TP criteria index presents in $ % and $ & at the same time; 3) add 3 if TP criteria index not presents in $ % and $ & ; 4) add 4 if $ %  W. For the GUESS method: 1) add 2 if criteria upper bounds are less or equal to lower bounds; 2) add 1 if criteria upper bounds are greater than nadir vector values; 3) add 1 if criteria lower bounds are less than ideal vector values; 4) add 2 if reference point values are not between nadir and ideal vectors values. 3.2

Testing problems Methods GUESS and STEM was tested with six testing problems: 3 2D and 3 3D, where Pareto set connected and Pareto front convex or concave. In all selected TP theoretic Pareto set is continuous. Problems was divided into four classes: A –UV ) continuous and concave, B – UV ) continuous and convex, C – UV ) nonsymetric and continuous, D – UV ) symetric and concave. Problem names noted TP1–BINH1 [16], TP2 –

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HANNE [17], TP3 – RENDON[15], TP4 – TAPPETA[18], TP5 – DOWNING[19], TP6 – VINNET[20]. See Table 1. All testing problems was reduced to single criterion optimization problem and solved with MATLAB (7.7.0.471) Optimization Toolbox function fmincon(). 3.3

Data analyses Obtained data was analyzed after experiment using descriptive statistics and frequency analyses of stopping reason. Data of each TP was grouped by stopping reason: 1) solutions fluctuate around the same value; 2) the goal was obtained. Then data was compared using descriptive statistics: mean, max, min, standard deviation and variance for all metrics described in previous subsection. Obtained solutions were graphically compared with theoretical Pareto front. Table 1. Testing problems definitions.

For all TP was obtained UV ) .

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4

Results and discussions

Experiment data was grouped by stopping reason for each TP and IOM. Table 2 shows frequencies of stopping reason for all IOM and TP, total TP solving repeat number – 10. Table 2. Stopping reason frequencies (analyzing last 4 iterations)

1. The goal was obtained

TP1 G S -

2. Euclidean distance increase

4

-

-

2

-

-

-

6

2

-

-

-

3. Solutions fluctuate around the same value 4. Solutions conv., to the goal slowly

1

10

-

8

4

10

10

2

-

10

-

5

-

-

-

-

1

-

-

-

-

-

-

-

5. Exceeded max iteration count

5

-

-

-

-

-

-

2

8

-

-

-

Stopping reason

TP2 G S 10 -

TP3 G S 5 -

TP4 G S -

TP5 G S -

TP6 G S 10 5

* G – GUESS method, S – STEM method. The goal solution by GUESS method was obtained (with predefined tolerance) solving TP2, TP3, and TP6. STEM method obtained the goal solution only solving TP6. Solutions fluctuates at the same value for GUESS solving: TP1, TP3 and TP4; STEM solving: TP1, TP2, TP3, TP4, TP5 and TP6. Table 3 shows results of case when solutions fluctuate around the same value and their metrics values. The metrics values are the same for all repeating times where dispersion and variance are equal to zero. Table 4 provides information of metrics values for cases when solutions fluctuate around the same value and the goal was obtained. The DMM solving TP4 by GUESS provided absolutely correct VRI information but values of upper and lower bounds of objectives and reference point values was small in all repeating times, this strategy is not appropriate for GUESS method. Table 3. Metric values are the same for all iterations (case – solutions fluctuate around the same value)

Method–Problem GD* MPE SP ED IC 0.0 0.0 0.0 329.02 3 GUESS–TP4 38.25 12.37 8.95 196.38 4 STEM–TP5 0.0 0.01 0.68 1.10 4 STEM–TP2 0.41 0.97 54.88 412.78 4 STEM–TP4 *see chapter 3.1 for legend The DMM solving TP5 by STEM was provided wrong VRI information (for all repeating times)– objectives indexes are repeated in the class $ % but the class $ & was mainly empty. The DMM solving problem TP2 provided empty class $ & but solving problem TP4 situation was similar to TP5. Table 4. Descriptive statistics for experiment metrics

Solutions fluctuate around the same value min GD MPE SP ED IC GD MPE SP ED IC

0.000 0.000 17.673 26.566 4 0.009 0.154 11.690 0.089 4

max

avg

X

STEM–TP1 (L =0.065), N=10 0.029 0.006 0.012 0.356 0.007 0.147 22.361 21.423 1.976 34.896 33.23 3.509 7 4.600 1.265 GUESS–TP3 (L =0.07), N=4 0.012 0.205 15.940 0.169 8

0.011 0.174 14.842 0.127 5.000

0.001 0.021 2.102 0.040 2.000

The goal was obtained

X

min

0.000 0.022 3.907 12.320 1.600

0.000 0.000 0.000 0.000 1

0.000 0.000 4.421 0.002 4.000

0.000 0.010 0.000 0.009 1

Y

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max

avg

X

GUESS–TP2 (L =0.1), N=10 0.000 0.000 0.000 0.070 0.036 0.023 0.000 0.000 0.000 0.070 0.036 0.023 1 1.000 0.000 GUESS–TP3 (L =0.07), N=5 0.014 0.244 20.357 0.040 5

0.006 0.122 10.096 0.025 2.600

0.007 0.108 9.477 0.014 1.817

XY

0.000 0.001 0.000 0.001 0.000 0.000 0.120 89.820 0.000 3.300

Solutions fluctuate around the same value min GD MPE SP ED IC GD MPE SP ED IC

0.000 0.000 11.262 1.471 4 0.012 0.191 1.185 0.170 4

max

avg

X

STEM–TP3 (L =0.07), N=10 0.006 0.001 0.002 0.173 0.031 0.065 17.877 16.754 2.412 3.799 3.447 0.789 9 4.800 1.751 STEM–TP6 (L =0.04), N=5 0.016 0.014 0.002 0.191 0.191 0.000 1.200 1.191 0.000 1.276 0.833 0.606 5 4.400 0.548

X

Y

The goal was obtained min

0.000 0.004 5.820 0.623 3.067

0.000 0.000 0.000 0.000 1

0.000 0.000 0.000 0.367 0.300

0.010 0.191 0.898 0.000 4

max

avg

X

GUESS–TP6( L =0.04), N=10 0.035 0.007 0.013 0.264 0.777 0.107 1.932 0.532 0.859 0.039 0.016 0.014 3 1.400 0.699 STEM–TP6 (L =0.04), N=5 0.014 0.012 0.001 0.247 0.201 0.025 1.236 1.105 0.139 0.008 0.003 0.004 8 5.20 1.643

XY

0.000 0.012 0.739 0.000 0.489 0.000 0.001 0.019 0.000 2.700

In a case when solutions fluctuate around the same value, the DMM solving TP1 by STEM mainly provided empty class $ & and objectives indexes not repeated in $ & . Solving TP3 by STEM the DMM mainly provided empty class $ & and repeated objectives indexes in the $ & . The DMM solving TP3 by GUESS was provided absolutely correct VRI values. In case when the goal was obtained the DMM solving TP6 by STEM mainly provided empty class $ & and objectives indexes repeated in class $ & . If the $ & class is empty then we can`t set upper bounds LZ for relaxed objectives and search process converge to the same solution every time. Analyzing values of general distance (GD) and maximal Pareto front error (MPE) we concluded that obtained solutions belongs to the UV ) or not. Less values near 0 are preferred, but a greater shows that obtained solutions are far from UV ) . GUESS and STEM methods as well as others IOM are sensitive of solutions + , belonging to UV ) , otherwise methods cannot obtain Pareto-optimal solutions, but only weakly Pareto-optimal. Spacing metric (SP) gives comprehensive of obtained solutions standard deviation. If SP value is near 0, then obtained solutions are very close each to other that is typical for solutions fluctuating around the same value. Otherwise SP shows that IOM has obtained different solutions (in different places of UV ) ). Figure 3 shows obtained solutions in Pareto front and Euclidean distance to the goal solution. Solutions in iterations 2 – 4 has equal values of objectives, the same situation for iteration 6 and 7 solutions. Provided preference information by the DMM was different for all iterations. The DMM obtained the goal solution very precisely (tolerance=0.004 and Euclidean distance equal to 0.0005) at iteration 8.

Initial solution Pareto front

Obtained solution

Goal solution

a) obtained solutions in Pareto front

b) Euclidean distance to the goal solution

Figure 3. The DMM solving TP6 by STEM (the goal was obtained)

Comparison of results showed that the DMM for GUESS method works better than STEM. On the other hand STEM method can`t bring close to the goal solution because values of VRI provided by the DMM was incorrect. In future to solve this problem we need to precisely define the second criterion of the (2) problem. - 57 -

5

Conclusions

The new DMM called ZuMo obtained goal solution of different TPs by GUESS method more times than by STEM. The goal solution by GUESS method was obtained solving TP2 (10 cases of 10), TP3 (5 of 10), and TP6 (10 of 10), while STEM only solving TP6 (5 of 10). Solving TPs by STEM the DMM many times provided wrong preference information (indexes of objectives are repeated in the class $ % ) for 3D problems. STEM method preference information rules allow keeping empty class $ & and not defining LZ – upper bounds of relaxed objectives. The DMM (2) problem second criterion fulfill the conditions of STEM method. Each IOM has its own best strategy for solving problems. For this reason criterion definition for complicated strategy will be difficult. For better solution results the definition of second criterion should be as precise as possible. Metrics values were taken into account to evaluate IOM effectiveness and the DMM behavior. Values of those metrics depends on how precisely UV ) solutions are calculated, not only on IOM effectiveness. ZuMo provides mathematical approach of IOM testing/comparison. Computation time can be reduced using parallel computation technologies. NSGA–II and others evolutionary algorithms has best opportunities on parallel computation. Researchers can design and implement experiments of testing/comparison IOM without human DMs reducing time and workload. ZuMo can be easy used for ad–hoc IOM comparison. The TP UV ) front and goal solution can be determined before the DMM start optimization.

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