knowcube for MCDM Visual and Interactive Support for Multicriteria Decision Making


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1 T. Hanne, H. L. Trinkaus knowcube for MCDM Visual and Interactive Support for Multicriteria Decision Making Berichte des Fraunhofer ITWM, Nr. 50 (2003)
2 FraunhoferInstitut für Techno und Wirtschaftsmathematik ITWM 2003 ISSN Bericht 50 (2003) Alle Rechte vorbehalten. Ohne ausdrückliche, schriftliche Gene h mi gung des Herausgebers ist es nicht gestattet, das Buch oder Teile daraus in irgendeiner Form durch Fotokopie, Mikrofilm oder andere Verfahren zu reproduzieren oder in eine für Maschinen, insbesondere Daten ver ar be i tungsanlagen, verwendbare Sprache zu übertragen. Dasselbe gilt für das Recht der öffentlichen Wiedergabe. Warennamen werden ohne Gewährleistung der freien Verwendbarkeit benutzt. Die Veröffentlichungen in der Berichtsreihe des Fraunhofer ITWM können bezogen werden über: FraunhoferInstitut für Techno und Wirtschaftsmathematik ITWM GottliebDaimlerStraße, Geb Kaiserslautern Telefon: +49 (0) 6 31/ Telefax: +49 (0) 6 31/ Internet:
3 Vorwort Das Tätigkeitsfeld des Fraunhofer Instituts für Techno und Wirt schafts ma the ma tik ITWM um fasst an wen dungs na he Grund la gen for schung, angewandte For schung so wie Be ra tung und kun den spe zi fi sche Lö sun gen auf allen Gebieten, die für Techno und Wirt schafts ma the ma tik be deut sam sind. In der Reihe»Berichte des Fraunhofer ITWM«soll die Arbeit des Instituts kon ti  nu ier lich ei ner interessierten Öf fent lich keit in Industrie, Wirtschaft und Wis sen  schaft vor ge stellt werden. Durch die enge Verzahnung mit dem Fachbereich Mathe ma tik der Uni ver si tät Kaiserslautern sowie durch zahlreiche Kooperationen mit in ter na ti o na len Institutionen und Hochschulen in den Bereichen Ausbildung und For schung ist ein gro ßes Potenzial für Forschungsberichte vorhanden. In die Bericht rei he sollen so wohl hervorragende Di plom und Projektarbeiten und Dis ser  ta ti o nen als auch For schungs be rich te der Institutsmitarbeiter und In s ti tuts gäs te zu ak tu el len Fragen der Techno und Wirtschaftsmathematik auf ge nom men werden. Darüberhinaus bietet die Reihe ein Forum für die Berichterstattung über die zahlrei chen Ko o pe ra ti ons pro jek te des Instituts mit Partnern aus Industrie und Wirtschaft. Berichterstattung heißt hier Dokumentation darüber, wie aktuelle Er geb nis se aus mathematischer For schungs und Entwicklungsarbeit in industrielle An wen dun gen und Softwareprodukte transferiert wer den, und wie umgekehrt Probleme der Praxis neue interessante mathematische Fragestellungen ge ne rie ren. Prof. Dr. Dieter PrätzelWolters Institutsleiter Kaiserslautern, im Juni 2001
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5 knowcube for MCDM Visual and Interactive Support for Multicriteria Decision Making Hans L. Trinkaus, Thomas Hanne Fraunhofer Institute for Industrial Mathematics (ITWM) Department of Optimization GottliebDaimlerStr Kaiserslautern Germany Abstract In this paper, we present a novel multicriteria decision support system (MCDSS), called knowcube, consisting of components for knowledge organization, generation, and navigation. Knowledge organization rests upon a database for managing qualitative and quantitative criteria, together with addon information. Knowledge generation serves filling the database via e.g. identification, optimization, classification or simulation. For finding needles in haycocks, the knowledge navigation component supports graphical database retrieval and interactive, goaloriented problem solving. Navigation helpers are, for instance, cascading criteria aggregations, modifiable metrics, ergonomic interfaces, and customizable visualizations. Examples from reallife projects, e.g. in industrial engineering and in the life sciences, illustrate the application of our MCDSS. Key words: Multicriteria decision making, knowledge management, decision support systems, visual interfaces, interactive navigation, reallife applications. 1 Introduction Usually, reallife decision problems are characterized by several criteria or objectives to be taken into consideration. Despite the fact that multiple objectives are all around us, as Zeleny (1982) points out, the dissemination of methods for multicriteria decision making (MCDM) into practice can still be regarded Corresponding author 1
6 knowcube for MCDM 2 as insufficient. A basic reason for that situation might be that practitioners are just not familiar with MCDM methods, which often are difficult to understand and to work at. One main difficulty for an easy and explorative usage of methods might be caused by the user interface, i.e. the handling of interactions with a corresponding computer program. Zionts (1999) is pointing out the necessity of userfriendly software in the following way: I strongly believe that what is needed is a spreadsheet type of method that will allow ordinary people to use MCDM methods for ordinary decisions. In this paper, we present a novel multicriteria decision support system (MCDSS), called knowcube, which mainly originates from focussing on userspecific needs. Therefore, the first aspect of the system is the provision of visualizations and interactivity in categories well understandable by nonexpert users. All technical stuff, necessary for this comfortable presentation of and access on knowledge, is hidden in the background, combined in a socalled navigation component. Its integration with further components, for generating and managing knowledge data required for decision processes, is another main feature of this MCDSS. The paper is organized as follows: Section 2 introduces some notation for multicriteria decision problems. (Here we restrict ourselves to tasks dealing only with quantitative criteria, other types of criteria will be topics of a paper to follow.) In Section 3, we present a survey of components of knowcube, and additionally some details on the generation and organization of decision knowledge. MCDMspecifics of knowcube, including a short excursion to human cognition, are discussed in Section 4. Two examples show the application of knowcube in reallife projects, one in the field of industrial engineering (Section 5), another one in medical radiation therapy (Section 6). The paper ends with some conclusions and an outlook to future work (Section 7), acknowledgements and references (Sections 8 and 9), and an appendix (Section10), meant to convince the sceptical reader of some ideas presented in the paper. 2 Multicriteria Decision Problems In the following we assume multicriteria decision problems (A, f) defined by a feasible set of alternatives A and a vectorvalued (q dimensional) objective function f =(f 1,..., f q ). For an alternative a A, f j (a),j {1,..., q}, represents the evaluation of the jth objective (also called criterion or attribute). Each objective function f j is assumed to be either maximized: max a A f j (a), or minimized: min a A f j (a).
7 knowcube for MCDM 3 If A is a finite set, i.e. A = {a 1,..., a r }, we also use the concise notation z ij = f j (a i ) fordenotingthejth criterion value for alternative a i. Here, Z =(z ij ) i=1..r,j=1..q is also denoted as decision matrix. In other cases, A may be a subset of a vector space, A R n, defined by a vectorvalued restriction function g : R n R m : A = {x R n : g(x) 0}. In practice, such a problem may be rather complex. For instance, restrictions defining the feasible set may be given by analytic formulas or differential equations. Usually, there does not exist a solution which optimizes all q objectives at the same time. Therefore, from a mathematical point of view, one may consider the set of efficient or Paretooptimal solutions as results to the above problem. For two solutions a, b A, wesaythata dominates b, iff(a) f(b) and { fj (a) f j (b), if f j is to be maximized, f j (a) f j (b), if f j is to be minimized, for each j {1,..., q}. The efficient set is then defined by E(A, f) :={a A : b A : b dominates a}. Since this efficient set normally contains too many solutions, its computation does not really solve the multicriteria decision problem from a practical point of view because a decision maker usually wants to get a single good solution, e.g. by choosing one from an easily to survey set of meaningful alternatives. For that purpose, an enormous number of methods for practically solving multicriteria decision problems has been developed in the meantime. Overviews and details referring to this are given, for instance, in Hanne (2001), Steuer (1986), Vincke (1992), or Zeleny (1982). One idea for example, common to some of these solution approaches, is to use additional preferencerelated information (stated by the decision maker) on the multicriteria problem, which then may result in obtaining a single solution. Popular concepts of these methods are taking specific values of the objectives (which may or may not correspond to feasible solutions) to compare solutions with. The earliest technique in MCDM using a desired solution ( the goal ) and minimizing the alternatives distance to it is goal programming, developed as an extension of linear programming. Today, there are many new methods which
8 knowcube for MCDM 4 utilize various kinds of desired and undesired solutions, different metrics, and additional features. Usually, these methods are called reference point approaches (Wierzbicki (1998)). In absence of userspecified reference points, the following two concepts are frequently used: The utopia (or ideal) point u is defined by the componentwise best values of the alternatives in objective space, i.e. { maxa A f u j = j (a), if f j is to be maximized, min a A f j (a), if f j is to be minimized, for j {1,..., q}. The nadir (or antiutopia) point v is defined by the componentwise worst values, but considering only efficient alternatives, i.e. for j {1,..., q}. { mina E(A,f) f v j = j (a), if f j is to be maximized, max a E(A,f) f j (a), if f j is to be minimized, Based on Simon s (1957) theory of satisfying, which was developed as an alternative to utility maximization, concepts using aspiration and reservation levels (or thresholds) became popular in multicriteria decision making. Taking aspiration levels for MCDM has first been pointed out by Tversky (1972a, 1972b). His approach of a successive elimination by aspects, is based on the idea that human decision making frequently works as a stochastic process for exploring alternatives. By choosing minimum requirements for the outcomes in specific criteria, the decision space A is reduced while the decision maker learns more about the feasible alternatives. Some features of the navigation process described below in this paper may be considered as an extension and userfriendly implementation of Tversky s basic concepts. Considering the limited capabilities of reallife decision makers, nowadays there is a significant number of methods providing ergonomic interactivity and means for visualization. Some of these are, for instance, described in Belton and Vickers (1993), Korhonen (1987, 1988), Korhonen and Laakso (1986a, 1986b), Korhonen and Wallenius (1988). A specific visualization technique used in our MCDSS, is called radar chart, spider web or spider chart. It is based on the idea of representing several axes in a starshaped fashion. An object characterized by multiple attributes then may be represented in the diagram as a polygon, resulting from connecting lines between the attribute scores on neighboring axes. Although such kinds of diagrams are rather common in statistics and brainstorming techniques, only few MCDM methods so far utilize this approach. And there are some additional new features in our MCDSS, as for example the fact, that the polygon representing an alternative by its objective values as corner points has got life : Online
9 knowcube for MCDM 5 mousemanipulations on the screen enable an interactive navigation through the database of decision alternatives. By talking so much about visualization, it should be appropriate just to do this as the sketch shown in Figure 1 perfectly illustrates the scope of knowcube: A decision maker, comfortably interacting (by ear, eye, mouth, or hand) with an MCDSS! black / white / hybrid multi criteria system Figure 1: One picture means more than thousand words. More explanations and especially practical details will be given in the next sections, here we only want to look still somewhat closer on the core of the multicriteria decision task, where we distinguish the following cases: If the description of the problem is known and completely given, it is called a white (box) multicriteria problem. This is the case mostly assumed in multicriteria optimization methods, applied for finding the best of all possible solutions. If nothing is known about structure of or dependencies within the multicriteria problem, it is denoted as black. Typically, outcomes of such systems are, for example, found by trial and error, or won by general inquiries. Frequently, such results are not reproducible, and in many cases all results are evolving in the sense that the problem settings are changing with time. Most real decision situations however are somewhere in between black and white, which lead to hybrid multicriteria problems. Then carrying out costly experiments and importing the results obtained into the database are the means chosen very often. But also simulation methods gain in increasing significance for getting decision suggestions. As already indicated in Section 1, here we are considering only quantitative criteria variables to keep it simple, to introduce the main concepts of knowcube,
10 knowcube for MCDM 6 as a first step, and to have place for explaining the basics by examples. For working with other types of criteria like qualitative, objective/subjective, rational/irrational, active/passive, dependent/independent, deterministic/statistic, hard/soft, timely,..., and all mixed together the topics introduced so far must be slightly extended. This will happen in a paper to follow. 3 Scope of knowcube, Knowledge Organization and Knowledge Generation knowcube, seen as a general framework of an MCDSS, consists of three main components: knoworg, knowgen, and knownav (knowledge organization, generation, and navigation). These are put together with their sub modules into a common box, showing to various decision makers varied views like the different faces of a cube. This gives a hint at the term knowledge cube, abbreviated knowcube, taken as a logo for the complete MCDSS. Figure 2 illustrates its structure. knowcube knoworg knowblo knowbas knowmet knowsto knowgen knowopt knowexp knowsim knowimp knownav knowvis knowsel knowdis Figure 2: The Structure of knowcube. The bricks shown there may act selfcontained up to a certain extent, but they also are designed for efficient interaction in the configuration of special
11 knowcube for MCDM 7 buildings for specific needs. So knowcube is customizable for use as a decision support system in various application domains. This will become more evident in Sections 5 and 6, where two examples are presented, one in the field of industrial engineering, another one from among medical applications. For an easy integration of knowcube into an institution s workflow, the component knoworg provides comfortable tools for data collection, recording, administration, and maintenance and all that with comparatively little effort. Open data interfaces enable information exchange with external data sources, such that the database of knowcube may be extended by arbitrary external, and also historic, decision support information. These topics mainly are handled by the modules knowbas (knowledge base) and knowsto (knowledge storage). (More details will be worked out in a future paper.) Making decisions presupposes some knowledge. So, in any case, a data base must be filled with decision information, where its fundamental unit is called knowblo (knowledge block). Such a block consists of two parts, one containing all those criteria values corresponding to a specific decision alternative (e.g. a row of the decision matrix Z, in case of a finite set of alternatives A), the other one references to additional information attachments. Those may be, for instance, documents, diagrams, graphics, videos, audios,..., any media are allowed. The usage of criteria and attachment information is best explained within application contexts, which will be shown later in the examples already mentioned. The last brick in the knoworg component to talk about is knowmet (knowledge metric). Here, the singular of the word metric is a bit misleading: There is not only one metric or one class of similar metrics defined on the data base, but additionally a big family of metric like orderings. For instance, all l p metrics (1 p ) may be used, at least for quantitative criteria blocks, together with superimposed priority rankings which furthermore are online user adaptive. Besides that, certain filtering functions allow some clusterings and inclusions or exclusions of sets of data blocks. So far for the first component of knowcube, corresponding to the data base symbol on the top in the sketch of Figure 1. There the dotted lines indicate, that the decision maker usually does not take care about these interrelations, they must act silently on the back stage. An engaged decision maker or, at least, the decision provider and analyst will show more interest in the second component knowgen (knowledge generation), which comprises the framework for producing decision information to fill the data base for a given multicriteria problem. Thereby, the main tasks are to formulate and answer questions such as e.g.: What kind of system will model the decision situation? Is the model static or does it change in course of time?
12 knowcube for MCDM 8 Which criteria are identifiable? How may the criteria variables be arranged in groups? Could information be aggregated following a topdown approach? Which addon media should be attached for supporting decision processes? After these preliminaries the interfaces between the acute knowledge generation domain and the components knoworg and knownav are adjusted. Then, one or more of the following activities may be started, corresponding to the bricks of knowgen in Figure 2: Optimization calculations (in the white system case) delivering e.g. Pareto solutions as outcomes of the module knowopt (knowledge optimization). Performing experiments (in the hybrid system case) observing results, e.g. from optimal experimental design methods, as produced by knowexp (knowledge experiments). Doing simulations (again in the hybrid system case) getting e.g. data records with a mixture of some hard/soft criteria, where the input and output values are generated by some standard simulation software tools, gathered in knowsim (knowledge simulation). Collecting data events (in the black system case) e.g. by reading information from external sources via knowimp (knowledge import). Using some other problem specific method which is not discussed above, and indicated in Figure 2 by dotted lines. In any case, the modules knowmet and knowsto are closely working together, if some newly generated or acquired decision knowledge (knowblo) is put into the data base (knowbas). The data base filling processes may be organized via batch procedures, interactively, and also in an evolving way, i.e. within several time slots and under various external conditions exactly as in real life, and also with the well known effect that old decisions are overtaken by new ones. As illustrated in Figure 1, knowledge generation happens mainly in the center of knowcube. So for example, by feeding or controlling the black/white/hybridsystem (possibly also by acoustic commands), and by receiving or observing corresponding system answers. A more distinct explanation of so far introduced concepts will be given below, especially in context with the two already mentioned application examples after a short break:
13 knowcube for MCDM 9 4 About Visual Data Processing and Human Cognition Towards knowcube Navigation Some simple tests (as given by the Appendix in Section 10) and thereof derived conclusions will motivate the design and functionalities of the visual interface of knowcube in general, and especially the ideas of the socalled knowcube Navigator. Let us summarize some basic principles of visual data processing which are well known long ago, and may be found e.g. in Cognitive Psychology by Wickelgren (1979) by three statements: Statement 1: Reading text information or listening to a speaker is not sufficient to generate knowledge for solving problems or making decisions! Statement 2: The human visual system has a highly efficient information processing architecture. It allows inferences to be drawn based on visual spatial relationships fast and with small demands on working space capacity! Statement 3: The human visual system is also highly sensitive for moving objects and changing shapes! Quintessence: Supporting decision making should appeal to the human visual system, transforming complex situations into timeanimated, spatial presentations! So, visual spatial relationships, moving objects and changing shapes all these general topics should be integrated in a tool for supporting decision making, at the least. But, additionally to that, such a tool asks for some more specific and practical demands on its man/machine interface. These requirements, gained by experience and evaluated by experiments, will be formulated in the sequel as axioms, and visualized by easy examples. The first step in the process of optical perception is the division of a picture into a figure, which comes by a filtering process to consciousness, and into the background, which is no longer of interest. The main question for designing visual interfaces based on figures is: What are the main principles for recognizing figures out of arbitrary forms? Many of the answers given below as axioms already have been discovered during the development of Gestalt Psychology, as a theory of perception by Wertheimer (1923) and others, during the 1910s and 20s. Nowadays, these ideas are common knowledge, and extensively used in multimedia applications; cf. Macromedia Director 7 (2000). Axiom 1: Closedness. The most elementary forms that are interpreted as figures are surrounded by closed curves.
14 knowcube for MCDM 10 Axiom 2: Closeness. Objects close together are seen as figures, distant objects are assigned to the background. An example is given in Figure 3. Each sketch contains eight domains, which are all closed. But they are hardly perceived all at the same time. The sketch on the left shows a cross standing upright, on the right it is inclined. Narrow parts are taken together as the arms of the crosses, broader parts within the sketches are seen as interspaces or background. (Only with a certain effort it is possible to discover an upright cross with broad arms in the sketch on the right, too.) Figure 3: Picture, Figure and Background. Now knowing the two basic principles of identifying figures: closedness and closeness, the question arises: Which principle dominates? Axiom 3: Closedness versus closeness. If closedness and closeness are competing, it s more likely that closed forms are taken as the figures looked for. Figure 4 should convince us: Broader domains become figures, because they are closed. The narrow sectors no longer appear as arms of a cross, since little parts of their boundaries are missing. Figure 4: Closedness versus Closeness.
15 knowcube for MCDM 11 Finally, to get closer to our goal: the knowcube Navigator, wepresentasa last concept needed: Axiom 4: Convexity. All convex forms or roughly convex forms as e.g. starlike shapes are particularly easy recognized as figures. This was already realized by v. Hornbostel (1922). Here is an example: Figure 5: Starlike Shape or Nearly Convexity. Though not looking quite nice, the polygon bounding a nearly convex domain is eyecatching. The line segments, oriented radially towards a common invisible center, are obtaining less attention. This contrast will still increase, if the line segments stay at rest, and the polygon starts to move. Such a polygonal representation is the most central visualization tool of knowcube s main component knownav, which provides the core navigation tools. So, we are getting closer now towards knowcube navigation: Each variable is related to a line segment, the line segments are arranged like a glory, each segment shows a scale (linear or something else), where small values always are put towards the center to avoid confusion. Each segment gets an arrowhead pointing towards that direction (bigger or smaller values), which is preferred by the decision maker for the corresponding criterion. Little circles on the segments represent the current parameter values of a data base record or decision alternative, respectively. Adjacent circles are connected by straight lines, all together defining a closed polygon, which bounds a convex or at least starshaped region. By this little trick the observer or decision maker comprehends what belongs together. Summing all up: A closed polygon within a starshaped scale arrangement visualizes a record in the data base knowbas. Figure 6 corresponds to a decision task, where eight criteria are involved.
16 knowcube for MCDM 12 Three of them are to be minimized, the other ones are to be maximized. (For the sake of simplicity, the scales on the segments are omitted here but they are shown in the second application example in Section 6.) The order of the criteria s arrangement may, for instance, reflect their importance, or it may be chosen interactively according to the user s preferences. Figure 6: A Decision Alternative, Represented in knowcube. To the end (of this Section) two questions still are open: How are information attachments supporting the decision making? What about the dynamic behavior of polygons? Best answers will be given by two already announced reallife applications: 5 Designing Best Trunking Devices We consider the industrial engineering problem of designing the profile of a new electrical trunking device that is usually fixed at the wall, and used for a comfortable laying of multitudes of cables. The problem is to find a best crosssection profile for such trunking devices. Many objectives are to be taken into account in such a development process. Among them there is one group of criteria, which should be minimized: a m, the total amount of material needed for one unit of the trunking device; t p, the time to produce one unit; c u, the overall cost for one unit. For a better understanding of a second group of variables, Figure 7 will help, showing in
17 knowcube for MCDM 13 front the approximately Ushaped crosssection of a trunking device. r p points to the resistance of the trunking device against pressing from opposite sides, r t to its resistance against torsion. These two variables should have big values, of course. Figure 7: A Trunking Device. The five criteria introduced so far may be considered as characterizing the outcome of a specific design for a trunking device. Their resulting values obviously are caused by a third group of (in a certain sense independent ) variables: t b denotes the thickness of the bottom face, t s the thickness of the side face of the device. A last variable p r (not visualized, but only mentioned in Figure 7) specifies the portion of recycling material, used e.g. for producing PVC trunking devices. All these eight variables are working together, in nontrivial, highly nonlinear, and partially conflicting combinations. Figure 8: Visualization of an Alternative.
18 knowcube for MCDM 14 Assuming now, that the engineering team has designed plenty of different profile versions, evaluated them virtually (e.g. by some finite element methods out of knowsim), according to the above criteria (which correspond to f 1,..., f 8, as introduced in Section 2), and put all together with some operational data into the data base, then some data blocks could be presented as shown in Figures Figure 9: Design Alternative 98. Figure 10: Design Alternative 746. With the generation of these data, the job of the engineering team is done. Now the time has come for the management of the enterprise to decide which new type of trunking devices is going to be launched. But, of course, a top manager would not want to look over hundreds of data sheets. Here the knowcube Navigator supports the user to stroll easily, quickly, and goal oriented through the data base. By pulling at a vertex of the navigation polygon, at a socalled grip point, towards smaller or larger values (inwards or outwards), the polygon moves or transforms to a neighboring polygon, representing another element of the data base. In the background the module knowmet is looking online for that new data record, which in the first place is different but closest to the old one, with respect to the criterion that was gripped, and which in the second place is
19 knowcube for MCDM 15 closest to the old record, with respect to all other criteria. It certainly would be an essential advantage for the top manager always to know where he/she is, and which goals are achievable in the best case. An additional graphical assistance will help here: For each criterion there are taken the respective values from all records in the data base. Virtually, the resulting point set is plotted on the corresponding line segment. The smallest interval containing such a set defines a decision range of potential alternatives for that specific criterion. Then connecting inner (neighboring) interval endpoints by straight lines determines the inner decision boundary, the outer decision boundary results by analogy. Both together construct the decision horizon. The metaphor horizon is used, because the decision maker knows and sees in advance that each possible alternative, respectively selection of a data record or corresponding polygon representation, will lie somewhere within this area that is emphasized in the sketch in Figure 11 by some color shading. (The utopia point u, mentioned in Section 2, corresponds to that polygon, which connects those boundary points of the shaded area, lying on the variables line segments and closest to their arrow heads. The nadir point v results vice versa.) Figure 11: What Is Achievable What Not?. By the way, the decision horizon serves as an ideal base for discussions at meetings, where many decision makers are fighting with plenty, usually conflicting, arguments. Acute viewpoints may be ranged in easily by the location of the navigation polygon, extreme positions may be estimated by the inner and outer boundaries. Alternatives, their pros and cons, are visualized interactively. The attachment knowledge, as e.g. the sketches of the cross sections in
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