Michal Radecký
email: michal.radecky@vsb.cz

Jan Martinovič
email: jan.martinovic@vsb.cz

Dušan Fedorčák
email: dusan.fedorcak@gmail.com

Radek Tomis

Ivo Vondrák
email: ivo.vondrak@vsb.cz

Using Extended Raster File for Real Time Traffic Information Mining

Gauging and analyzing the ever-growing strain on todays roadways is an issue that needs to be adressed with urgency. The rate at which technology is developing and the amount of vehicles now equipped with GPS systems are factors that ensure the potential for creating statistics to gauge this strain. When processing data gathered from monitored vehicles, it is necessary to implement procedures that identify specific roadways upon which a given vehicles movement is recorded. With respects to the volume of this data, a method for indexing these data files becomes a critical issue. Within the following text, we will present a process of collecting and processing data in the FLOREON+ Traffic system and spacial indexing using a raster index that processes queries at a much greater speed than standard indexing.

Introduction

The problem with analyzing the increasing strain on today's roadways lies within the fact that monitoring is limited to major intersections and trouble spots, where modern camera systems have become the norm. Publicizing information gathered from these areas leads to faster solutions in crisis situations and aids in the overall elimination of such crises. A great disadvantage in this approach, however, is the fact that a crisis situation needs to be identified in advance in order for it to be resolved in time. A system based on collecting and processing data derived from a vehicle in real time may contribute to the exploitation of this disadvantage -the ability to monitor only vehicles where this equipment is installed. Currently, the amount of vehicles thus equipped is too large for a system of this type to be of any significance or to even be successfully implemented.

Floreon + is a science-research project aimed at building a prototype system for modeling, simulating and monitoring situations caused by unfavorable, natural phenomena using modern computer and Internet technology. The main feature of this project is its focus in the area of crisis management and support. The project has gradually expanded to cover relevant topical fields, such as flash flooding, and other factors pertinent to crisis management and general information services.

Recently, traffic problems have become a recurring issue to be dealt with on a daily basis in various forms and at several levels. Aside from the everyday traffic situations perceived by drivers, crisis situations, management and solutions must be taken into account. This is precisely why the field of traffic information and management are intricate parts of the Floreon + project. Knowledge of traffic situations and the ability to utilize sophisticated apparatus to effectively process this information are significant factors for an entire line of activities from the perspective of crisis management (e.g. which traffic lanes are clear for emergency unit passage, which areas are flood hazards, etc.). Additional benefits would obviously include the acquisition and processing of everyday traffic situations and activities (intelligent navigation systems for automobiles in metro areas, analyzing high-traffic zones, detecting problems on roadways, etc.)

The Floreon + Traffic system makes calculations (in real time or for potential situations) by utilizing a complete line of information derived from various sources. Data acquired in this manner is then analyzed and adjusted to fit a format that is then usable in the aggregation of this data with connection to time and space. Currently, the main information source remains the vehicle itself. Thanks to our cooperation with companies involved in monitoring car parks, we are able to acquire surface data and individual car speeds in real time. Individual data acquired in this way is anonymous enough to fulfill its purpose. Once received, this information is combined with other data relevant to a given traffic area at a given time. The system actually works with data to determine an automobiles' speed and driving style.

Navigation systems

Nowadays, the modern technology, especially information technology, is one of the main parts of automotive industry. There are many new features which bring new driving experiences and encourage driver's skills, e.g. adaptive cruise control, lane assistant or traffic signs reader. However, the on-board navigation is still most common tool for daily personal transportation.

For today's drivers, it is very useful and required to know the route from the one place to another. Unfortunately, the simple knowledge of static route itinerary is not enough. There are many factors which affect the travel time and efficiency of the selected road. These factors are changed during the time and they are also depending on time. So, it is necessary to integrate dynamic models of traffic within the routing approaches to offer real-life navigation functionality. These dynamic models can cover real behaviors of traffic on macro level based on time, knowledge of statistical data related to infrastructure elements, knowledge of real traffic occupancy and utilization, etc. Based on these, it is able to develop much more sophisticated routing algorithms that bring whole new perspective on the on-board navigation systems. But of course, on-board navigation is not only one 'consumer' of such approaches and theirs results. Also the traffic management or logistics business are covetous users of that.

The crucial layer of these approaches is data layer. The ability to collect relevant data and its preprocessing are the very important parts. It is hard to develop dynamic models without suitable data related to the time and location. The information on streets capacity, its utilization, traffic restrictions and incidents, traffic jams, weather, etc. is necessary to have as a background for dynamic model development and operation. Some of these data (GPS positions with speed value) are possible to collect based on cooperation with car fleet providers, however these data are often geographically and quantitative limited. The companies that develop and offer navigation solutions are able to solve these problems. For example, the TomTom company offers the IQ Routes technology 1 , that deals with standard on-board navigation devices and theirs users which participate on global traffic state knowledge. This solution puts, by online or offline ways, the driving experience of millions of TomTom users into central knowledge base and calculating routes based on current day time, actual speeds driven on roads compared to speed limits.

The future of navigation and other related tools and services is covered by integration of all accessible data sources and its utilization within complex traffic model. On this background, there is able to offer sophisticated 2 Floreon + Traffic

We are currently receiving information on approx. 2,200 vehicles of different types (passenger cars, cargo trucks, lorries, etc.). This data is aggregated into a simple, speed fast profile in real time around the entire Czech Republic. (Coverage is only dependent upon the accessibility of vector data. Acquired data is currently limited to identifying major roadways because an application that covers all streets would not be relevant enough with respect to the amount of moving vehicles and the vastness of this network.) In the near future, we plan to carry out much more detailed analyses of this data in a way that allows us to randomly expand the grid structure of intersections. This expansion will aid in creating a necessary base of information for evaluating operations and applications of graph algorithms. Currently, the system is receiving anonymous records from approx. 2,200 vehicles. During rush hour, that equates to about 13 records received/second. Taking into account the algorithm being applied, unprecise GPS readings and, specifically, relevant vehicle data (unignited engines, etc.) the success rate for the placement of given information on a specific location is around 30%.

This information is generally highly relevant to the value of real time situations on roadways. Nonetheless, due to the need for high quality and a large quantity of acquired data, this information must be supported with data derived from alternate sources. The most significant data acquired is, therefore, that which is acquired directly from a given infrastructure. Such data may be derived from toll stations, telematic sensors, camera systems, etc. Within the scope of their given locations, these sources may influence aggregated data derived from a specific roadway. The advantage provided by these data sources is the high level of precise information they provide (amount of vehicles passed, etc.). On the other hand, the disadvantages of using these tools as data sources are their predominately localized scope and their high investment and operational costs. However, an appropriate combination of data acquired on-line and data acquired directly form infrastructures allows for the creation of a quality image of the overall traffic situation.

An equally important data source would be that of an agenda system that provides information on traffic accidents, road closures, weather, etc. The potential of this alter- native as a valuable source is currently being developed in cooperation with operators of the JSDI system (A unified traffic information system in the Czech Republic).

Floreon + Traffic system (scheme in Figure 1) output may be assessed and defined in different ways. Emphasis, however, is placed on its future potential. Aside from providing aggregated data for additional processing, we envision future benefits of this system in the following outputs, applications and expansions:

-Detailed statistical information for individual areas, streets, and roadways.

-Accountability for evaluating roads according to amount, quality and data source influenced by a given road. -Visualization of entire lines of communication (camera systems, traffic accidents, data quality, etc.) in 2D and 3D imagery. -Provision of functions and operations on data (searching for optimal roads, etc.).

-Processing and integrating data derived from various data sources (JSDI, telematic equipment, meteo. data, digital images, etc.). -Analysis of historical data with current situational perspectives (automatic detection of problems on roadways based on comparisons of normal speeds and actual speeds on a given road).

-Application of higher level algorithms (e.g. anthill theory, flood wave algorithms) to create predictions and simulations usable in other systems, primarily from a traffic management perspective.

The applicability of this entire system lies directly in its dependence on data, its quality, quantity and heterogeneousness. The advantage in using data acquired directly from vehicles is in their surface coverage and immediate accessibility, regardless of their lack of relevance. Once this approach is combined with data derived from telematic sensors, for example (limited to local placement, but highly reliable and precise) it will be possible to create a total and real traffic situation model.

Collecting and Processing Data

As mentioned above, the project we are dealing with is based on collecting and processing data, derived directly from automobiles equipped with GPS functions that are able to send spacial data to a distant server in real time, where it is to be processed. The process then becomes a joint attempt at identifying a roadway where a vehicle is in motion. Using this identification, we place a spacial query above the roadway network. From data prepared in this manner, it becomes possible to effectively set statistics for roadway strain and appropriately visualize their results.

Raw data derived from a vehicle is sent at an amply high rate, i.e. at an average of 1 vehicle/10sec. With an adequate set of monitored vehicles (in our case, we have calculated with 20,000 vehicles), we attain the desired speed of 2,000 recordings/sec. Data derived from vehicles obviously does not flow at a constant rate. This issue, however, may be easily resolved with an appropriately-sized memory. One question remains, however, as to whether it is at all necessary to process raw data in real time. If we perform a rough estimate of memory complexity, say for a two-month interval of a monitored traffic situation, for a recorded surface and a vehicle's speed, we quickly reach hundreds of gigabytes of stored data. With data processed from an identified roadway, upon which their is vehicular movement, we achieve a much lower volume of data (this obviously depends on the strain of traffic on the given network of roads being processed).

Another factor is the speed at which data is processed. Whether we process data in real time or in batches, identifying roadways where there is vehicular movement is both time consuming and technologically demanding. We will also need to factor in several potential difficulties that will likely arise, of which the most significant may be GPS precision. For this, it will be necessary to carefully install error intervals to aid in partially eliminating this problem.

Another issue is the actual size of the roadway data file itself. In order to provide meaningful statistics, it will be necessary to include second class roadways. A file this size for the entire Czech Republic will be quite large and it will have to be queried for every processed recording from every vehicle being monitored. If we use the Post-greSQL tool2 as a reference database, with PostGIS3 space expansion, and the commonly used GiST index, the speed a spacial query is sent to the roadway database will still be dependent upon the size of the data file itself. As described above, this data file is quite large and it is almost impossible to reach process speeds greater than 2,000 queries/sec. For this reason we have decided to develop our own method of indexing, based on scanning domains, making time complexity independent of the amount of indexed data.

Raster Indexing Spacial Data

The principle of raster indexing, in and of itself, is quite simple. In several aspects, it is very similar to bitmap index. Simply put, a spacial raster index is a very big picture where all indexed recordings are drawn (geometry). Instead of using colors, recordings are "drawn" with an identifier. We can then effectively direct queries to this type of index (we can query the "color" of a specific pixel). The complexity of this type of query will obviously be constant. One issue with raster indexing is the limited amount of whole-number attributes upon which the index is performed. We must be able to clearly name all values of a given attribute in order to be able to use it for indexing. The raster index will then contain a paired key-value, where the keys are all value combinations of an indexed attribute and and the value is the recording identifier that answers to a combination of attribute values. The index file itself does not contain a key because it is already possible to encode a key to a position within the index file. A raster index defined in this way has one evident disadvantage: it is not possible to index two recordings with identical index attribute combinations. A solution for this shortcoming is described below.

As far as spacial data is concerned, the whole-number issue presents quite a significant problem. This problem is, however, resolvable. We must set an area border for the raster index we are creating and the resolution with which we will be working. Here, we come up against one troublesome aspect of raster indexing: with resolution comes inaccuracy. When creating an index, it is necessary to consider the application for which the index is being used and, based on this, set the adequate resolution.

Constant complexity, for which point queries are carried out, is attained with a massively increased complex memory. For the project of measuring roadway strain, we have chosen a spacial index resolution of 5m/pixel. Since a raster index is actually a discretely modeled, spacial index data domain, the size of the index created surpasses the size of the data file many times over. Aside from the amount of space used up in the discs storage, this issue does not present any additional complications. We must, however, factor in the disadvantage of occupied space in the discs storage. Since the data file itself does not change, the index file does not change either. If only the data file changes, however, it still is not necessary to go through the entire index file; just going through the space within which the data file appeared is sufficient. The interval in the index file which needs to be renewed is again identifiable with constant complexity. Creating, querying and maintaining a raster index for spacial data is subject to several other rules and conditions, to be described in a later section.

Raster Index Construction

As described above, we have constructed a spacial index for analyzing roadway strain that features the following properties: the size of the indexed space has been chosen so that the entire Czech Republic is covered and the resolution has been set at 5m/pixel. The structure of the index file, in its basic form, is as follows: the file contains a heading with a configured index (see Table 1) followed by a frame saved in the identifier recording. Every frame contains a matrix of 1024 × 1024 identifiers (see Figure 2). The frames are saved in the index file in rows.

The indexed data file itself is performed as follows: we gradually go through the space defined by inputting intervals and queries as commonly done with spacial databases (with a resolution of 5m/pixel, that equates to a square with a 5,120m edge) and we draw (raster) the resulting object onto a tile. Tiles are gradually saved in the file. The spacial point query for this type of index then simply lies in the index file based on its assigned coordinate. This results in the identifier object being located on the squares edge, measuring 5m (index resolution), despite its assigned coordinate being in the center of the queried square.

A spacial index constructed in this way, however, has several negative properties. For example, it is impossible to record more objects in one cell, which may present a big problem in the case of roadways. Another problem is the size and effectivity of the saved frame within the index file. Methods for countering these negative properties will be described in the following chapter.

Raster Index Optimizing

The first problem discussed, saving more files in one cell, can be easily solved by expanding cell recordings. In place of an identifier, we are going to store a paired-sign identifier. Within the one-byte field of signs, we can save information clarifying whether this position is just one object or if there are more objects present. Within the scope of a one-byte field signal, we can save information on whether this position is occupied by either one or more objects. If there is more than one object present, the object identifier is not saved here; its position is saved with the aid of a file where the situation is recorded in a query space (e.g. 3 objects with their identifiers). Another option for expanding cells saved on the frame is very closely related to indexing roadways. One valuable piece of information useful in expanding cells is the position of a queried point for a given roadway (defined as the distance from the start of the roadway, and running parallel to the entire queried point). Knowing this information enables us to effectively analyze and save data about a strained roadway without having to separate it geometrically into smaller segments. Here, It is important to consider the fact that every expanded cell within the recording increases the-overall index file size. If we had an identifier saved as a 4-byte whole number, expanded by a sign byte and the road position (4 bytes, float), our index file size increases by more than half its original size. This is best possible option for optimizing memory complexity.

By performing simple statistical calculations on an index file, we find that in the case of an indexed roadway (generally all LineString data types), the final index file is more than 90% "empty". The aim of lowering memory complexity will then depend on our ability to avoid saving empty space in the index, if possible. The solution to this problem is to build a quadtree [START_REF] Samet | Foundations of Multidimensional and Metric Data Structures (The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling[END_REF] above the matrix frame that is saved in the index file. All nodes in the quadtree will then contain a sign informing us whether or not the frame below it is empty. If not, every leaf contains a frame position in the index file (see Figure 3). Implementing a quadtree actually increases the time complexity of a queried index, but this complexity is dependent on the size of the queried space, not on the amount of indexed data. If we have a raster index the size of n pixels, we will need log n layers in order to get to only one tree root. A certain disadvantage here may be the square index space and limitations in the size of the line value 2n.

We do not need go all the way down to the root of the tree, though. We can just set an index from the matrix of the quadtree, which will allow us to partially eliminate the aforementioned shortcoming.

The next and last optimization for the system stems from the input of monitored traffic situations using GPS. GPS alone is able to determine a vehicles position within approx. 5m. Using maps to identify roadways with vehicular movement is also, at times, inaccurate; which is why the resulting overall error is added to the identification of a roadway error. This problem is well-resolved by defining the width of a road before rerastering its geometries in the index file. Paralleling this example would be like drawing a fatter line over a fine line using a bigger marker. An index created in this manner is able to smooth out smaller errors, even though it may cause further issues, especially when two rows lie very near one another (e.g. a highway that has directions recorded as two differing geometries). This shortcoming may be eliminated by recording perpendicular distances from the center of the roadway, again, at the cost of increasing the index files size.

Result from System FLOREON + Traffic

We are currently receiving information on approx. 2,200 vehicles of different types (passenger cars, cargo trucks, lorries, etc.). This data is aggregated into a simple, speedfast profile in real time around the entire Czech Republic. The overall coverage (with speed profile visualization) of the Czech Republic by floating cars used in FLOREON + Traffic system is on the Figure 4 . The above mentioned approaches form a background for a whole operation of traffic data processing. It is performed by FLOREON + system on the data storage layer, data mining layer, as well as visualization of results layer. It is important to note that the real traffic data from vehicles is not only source of information important for building traffic model and profiles. Also agenda data (traffic conditions and limitations, etc.), weather data, infrastructure data, live-cam data, etc. are important for that. The integration and combination of a wide set of data sources provide a real power to create effective and usable traffic model. So, this system is responsible for receiving the data from its providers (fleet car provider, provider of traffic infrastructure state, weather provider, etc.), aggregating the data based on time division and the location of a given records, preparing and performing the algorithms working with a wide range of data, concentrating all of these to the traffic model on micro, meso or macro levels and also providing simulation based on this model, preparing the outputs for following utilizations or integration to other systems.

The Figure 5 and Figure 6 depict a set of graphical outputs based on data processed by the FLOREON + Traffic system. Also, the traffic visualization is a part of functionality of general FLOREON + web interface 4 which is developed with emphasis to user experiences. The Microsoft Silverlight technology 5 is used for that. The next figure is a screenshot taken from this our rich internet application. The main pitfalls of this approach is the primary dependence on the floating car data [START_REF] Schfer | A traffic information system by means of real-time floating-car data[END_REF][START_REF] Sohr | Quality of Floating Car Data[END_REF], their quality and quantity. Amount of input data can be significantly reduced during the processing. Some data has not sufficiently informative value or are geographically inaccurate. This index based approach, where the individual anonymised records are assigned to a road, is loss (some amount of data is useless due to above mentioned reasons). The elementary knowledge of the motion vector of data sources would be an important contribution to the efficiency of data processing. Thanks to this, it should be possible to process not only one single location, but also the direction of the motion. It allows more complex traffic data processing. Unfortunately, this approach is unfeasible due to law on anonymity and data protection.

Conclusion

We presented the method of fast querying to the map of the road network based on the raster index. The method was experimentally and operationally verified by running Fig. 6. A different type of floating cars data visualization is shown here. The anonymous locations and speeds are placed on the map and create basis for indexing algorithms. Not all of these 'circles' involve the speed limit of particular road due to infrastructure limitations and GPS inaccuracy.

FLOREON + for three years, receiving information about approx. 2,200 vehicles every day. Obtained and aggregated data will be used for dynamic routing, which means data for routing will be updated accordingly to average situation on a given road section during day. In future work, we will be implementing this to our own version of routing, which is the combination of Highway Hierarchies [START_REF] Sanders | Highway Hierarchies Hasten Exact Shortest Path Queries[END_REF][START_REF] Sanders | Engineering Highway Hierarchies[END_REF] and heuristics [START_REF] Hart | A Formal Basis for the Heuristic Determination of Minimum Cost Paths[END_REF][START_REF] Goldberg | Computing the shortest path: A* meets graph theory[END_REF] and also to our approach of road network updating during accidents. We presented earlier in [START_REF] Kromer | Ant Colony Inspired Algorithm for Adaptive Traffic Routing[END_REF]. Next step is preparation of other types of indexes for routing. Currently, we are creating the index for routing algorithms based on OpenStreetMap data6 , which will also be used for optimized algorithms working on these data. There are sometimes incorrectly stated speed limits in data, so it's useful to replace them with average speeds on roads received by previously described method directly from vehicles.

1

  TomTom: http://www.tomtom.com skutečná vozidla vybavená GPS a datovou konektivitou zařízení poskytovatelů dat, která zpracovávají data od vozidel a poskytují je anonymizovaně dále datová úložiště systému Floreon+ obsahující agregovaná data a další zpracované informace zpracování dat z různých zdrojů -jádro systému Floreon+ Tra c systémy poskytující data -agendová, telemetrická, geogra cká, atd. a set of real cars with GPS and data transfer capabilities data providers servers that process the real tra c data and provide them in anonymized form Floreon+ data storages include agregated data and other processed information third-part systems interpretation and visualization of Floreon+ data processing of data from a several sources the Core of Floreon+ Tra c third-part systems that provide data -agendas, telemetrics, geographical, etc.

Fig. 1 .

 1 Fig. 1. Floreon + Traffic

Table 1 .Fig. 2 .

 12 Fig. 2. Structure of the index file

Fig. 3 .

 3 Fig. 3. Quadtree -two gray leafs are pointing at frame position in the index file

Fig. 4 .

 4 Fig. 4. The overall coverage of FLOREON + data with speed profile visualization

Fig. 5 .

 5 Fig. 5. Map visualization of speed profiles on the infrastructure (Ostrava, Czech Republic). The information on traffic conditions (roadwork, weather, etc.) is also included, as well as real-time camera view.

PostgreSQL: http://www.postgresql.org

PostGIS: http://postgis.refractions.net

FLOREON + : http://www.floreon.eu

Silverlight: http://www.silverlight.net

OpenStreetMap: http://www.openstreetmap.org

Acknowledgment

This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).


Pavla Dráždilová
email: pavla.drazdilova@vsb.cz

Alisa Babskova
email: alisa.babskova.st@vsb.cz

Jan Martinovič
email: jan.martinovic@vsb.cz

Kateřina Slaninová
email: katerina.slaninova@vsb.cz

Štěpán Minks

Method for Identification of Suitable Persons in Collaborators' Networks

Finding and recommendation of suitable persons based on their characteristics in social or collaboration networks is still a big challenge. The purpose of this paper is to discover and recommend suitable persons or whole community within a developers' network. The experiments were realized on the data collection of specialized web portal used for collaboration of developers -Codeplex.com. Users registered on this portal can participate in multiple projects, discussions, adding and sharing source codes or documentations, issue a release, etc. In the paper we deal with strength extraction between the developers based on their association with selected terms. We have used the approach for extraction of initial metadata, and we have used modified Jaccard coefficient for description of the strength of relations between developers. Proposed method is usable for creation of derived collaborators' subnetwork, where as input is used the set of words, which will describe the area or sphere, wherein we want to find or recommend suitable community and the words specify relation between the developers in the network. Obtained subnetwork describe a structure of developers' collaboration on projects, described by selected term.

Introduction

Recently the concept of social networks and online communities is becoming still more and more popular. As a result, the number of their users significantly increasing. Reasons for communication between people and creation of social networks in our time are various: study, hobby, work, games and programming is not the exception.

OSS (Open Source Software) is a example of a dynamic network, as well as a prototype of complex networks emerging on the Internet. By working through the Internet, interactions between developers can be considered as relations in the synthetic network of collaborators. These relations arise when the developers join the project and begin to communicate with others. OSS network consists of two entities -developers and projects. An examples of such OSS social network established on the basis of interaction between the participants is CodePlex.

Many programmers on the Internet are looking for interesting ideas, or assistance when implementing their own solutions. Online collaboration is no longer a novelty in our times and it is run by people all over the world. However, searching for suitable and capable people who could implement a particular idea at reasonable deadlines and high quality is an eternal problem.

In this paper we try to determine the strength of relationship or similarity between CodePlex developers in the context of projects they work on. To determine the context, we used project key words, which in the case of the CodePlex are extracted from project descriptions. We would find some developers or some community, which is specified by key words, for a recommendation.

Some related work dealing with the recommendation in the social network. In the article [START_REF] Chen | Make new friends, but keep the old[END_REF] authors studies people recommendations designed to help users find known, offline contacts and discover new friends on social networking sites. Other approach is in the article [START_REF] Long | Social network structures in open source software development teams[END_REF], where authors examine the dynamics of social network structures in Open Source Software teams but data were extracted monthly from the bug tracking system in order to achieve a longitudinal view of the interaction pattern of each project. Generally, registered users can participate in multiple projects, discussions, adding the source code and documentation, issue a release, etc. Some of the users have defined a specific role within the project for which they work. Each user has his own page, where he can share information about himself, his projects on which he currently works, and the most recent activities. The CodePlex projects themselves can be considered as a very interesting source of information. In addition to the list of users and roles, Code-Plex enables register keywords, add description of the project, the number of visits, status, date of creation, url and other information about the project. All activities are carried out on CodePlex by a particular user within a specific project.

CODEPLEX

Database which was created as a result of data obtained from CodePlex.com, consists of 6 main tables: User, Project, Discussions, RecentActivity, Membership and SourceCode (see Table 1).

In CodePlex, we can see two types of entities: users and projects. Both are represented by tables that contain specific characteristics. The table User contains informations about users such as login, personalStatement, createdOn, lastVisit and url of user page. The table Project contains some characteristics of project in Codeplex: tags, date od created on, status, license, pageViews, count of visits, description and url of project page.

The undirect connection between the user and the project is implemented through activities within the scope of the project. These activities are in the database CodePlex We can represent CodePlex as a bipartite graph of users and projects, where the edge between the user and the project is a user's activity in a project.

If we look at the data that we have in the Table User, we are not able to define the user's profile. It consists of the field of interest, what he deals with, the programming language he uses and at what level. PersonalStatement attribute is used to describe the user, but from the total set of our users downloaded, there was not a single one, who would fill it up. On the other hand, the project has enough information defined -which fields are concerned, how long it lasted, whether it is completed, which technology it is used, etc.

The main attribute, carrying the largest set of information, is the project Description -the description of the project itself.

Using activities such as user links to the projects, we are able to determine with some probability an area of specialization and a work of each user. For example, if a user is working on three projects written in .NET and one in Java, we could include him in .NET programmers with high probability, and less likely recommend him as a Java programmer.

In other words, terms or description of the project may not only help us to provide more information about projects, but also to determine the user's area of interests or abilities. As a result, the way we are able to compare user attributes determines the similarity to other network participants.

Whenever we think about collaboration between two persons, we not only look at the relationship itself, but also at the context. It is clear that depending on context, the strength of relationship changes. Therefore, we divide collaboration into two main parts Developers' Relationship and Developers' Context. We consider the relation between developers and the term describes the context between developers.

Developers have additional attributes. Usually it could be publications, teams, organizations, projects, etc. We called it attribute domain, in our case D CP D be a set of projects in Codeplex, then CP D are attributes for all D i developers, where objects is one developer's attributes described as CP D i ⊆ D CP D .

Developers' and Context Relationship

We describe a developers' relationship as commutative operation on cartesian product of developer's attribute X x X, where output is mapped to the set of real numbers R.

We use Jaccard coefficient ( [START_REF] Deza | Dictionary of Distances[END_REF] for evaluation of developers relations using their attributes.

Attribute Score (CP D i ,CP D j ) = |CP D i ∩ CP D j | |CP D i ∪ CP D j | (1)
As we discussed above, every developer has it's attributes. Moreover, each project has a description text. If we use lexical analysis on this text, we can define a term set for every developer as T D i and this term set contains all terms of projects, which developer D i participated. The extracted text is proceed to methods, which remove words that do not carry any important information. The main issue of this paper is not to describe this kind of methods. More could be found in [START_REF] Porter | An algorithm for suffix stripping[END_REF][START_REF] Konchady | Text Mining Application Programming[END_REF].

Term set T consists of all developers term sets {T D 0 , T D 1 , . . . , T D n } = T , when the domain for terms T could be obtained as union of all terms extracted for each person

D T = T D 0 ∪ T D 1 ∪ . . . ∪ T D n .
The whole process of obtaining term sets is described in [START_REF] Minks | Author cooperation based on terms of article titles from dblp[END_REF], so we just remiding (t k in T D i ) stands for the number of terms t k by T D i and (t k in T ) stands for the number of terms t k in descriptions of all projects by T .

We can evaluate association between the selected term t k ∈ D T and a developer

D i ∈ D: R(T D i ,t k ) = (t k in T D i ) (t k in T ) + |T D i | -(t k in T D i ) (2)
We normalize R(

T D i ,t k ) such that R Norm (T D i ,t k ) ∈< 0, 1 >: R Norm (T D i ,t k ) = R(T D i ,t k ) MAX(R(T D i ,t 1 ), . . . , R(T D i ,t |T D i | )) (3)
Evaluation of the whole relationship context of two persons D i and D j has two steps. First, we compute association between D i and select term t k , and between the second developer D j and t k separately. Afterwards, because each part is already evaluated by real number, we combine both results in the same way; we can combine the whole result in equation one. In CodePlex we see the description text for the developer as the all description of all projects he is working on, joined together. We obtain equation for the Context Score :

Context Score (T D i , T D j ,t k ) = R Norm (T D i ,t k ) R Norm (T D j ,t k ) (4)

Collaboration -Whole Score

The last step is to define Score, which consists of Attribute Score and Context Score :

Score(CP D i ,CP D j , T D i , T D j ,t k ) = Attribute Score (CP D i ,CP D j )Context Score (T D i , T D j ,t k ) (5)
This equation evaluates the relation between developers depending on the selected words, which represent the context. So we get a evaluation for the new subnet, which is specified by selected terms.

Construction of the Collaborators Graph

To describe the network of collaboration, we use standard weighted graph G D (V D , E D ), where weighted function is defined as w D : E D (G) → R, when w D (e) ≥ 0.

The determination of set D is simple, because objects of vertices set V D match with objects of set D, so V D = D. However, we can do the same with all the possible pairs from set D to assign a set of edges E D ; it is better to design the algorithm to each implementation at first, and to reduce the number of useless computations. In addition, we must choose term t k for function w D , which reflects the context. Because only the commutative operations are used, we do not need to take into consideration the order of attribute objects in function parameters. Moreover E D is two-object set, where the order of objects does not matter, so the evaluating is done just once.

When we construct graph based on developers' projects relationship, we use Attribute Score (CP D i ,CP D j ) as w D , where no term is needed, then simply V D = D, which means that every developer is a vertex in the graph. Then, for each developer D i ∈ D we find collaborators D i C and for each collaborator D j ∈ D i C we create two-object set {D i , D j }, which corresponds with an edge in the graph. Equation 1 is then used to evaluate the edge.

The function Score(CP D i ,CP D j , T D i , T D j ,t k ) is used for evaluating the edges in the context of the term. The only difference is, that majority of developers has not chosen term in their description text, so the result will be 0 and no edge would exists. Hence, we first determine subset of developers D t k ⊆ D for those that have a term in their description text, followed by the same steps described in the last paragraph to compute developers' projects relationship. Then, the term t k is used for computation of the second part in Context Score (T D i , T D j ,t k ). Finally, we calculate the whole Score by multiplication of both parts.

Construction of the Projects Graph

We consider as well as developers, as well as projects. We define Projects' Relationship and Projects' Context.

We use Jaccard coefficient for evaluation of projects relations using their attributes -Attribute Score .

Attribute Score (CP P i ,CP P j ) =

|CP P i ∩ CP P j | |CP P i ∪ CP P j | (6)
We evaluate association between the selected term t k ∈ D T and a project P i ∈ P:

R(T P i ,t k ) = (t k in T P i ) (t k in T ) + |T P i | -(t k in T P i ) (7)
We compute equation for the Context Score :

Context Score (T P i , T P j ,t k ) = R Norm (T P i ,t k ) R Norm (T P j ,t k ) (8)
The last step is to calculate Score, which consists of Attribute Score and Context Score :

Score(CP P i ,CP P j , T P i , T P j ,t k ) = Attribute Score (CP P i ,CP P j )Context Score (T P i , T P j ,t k )

To describe the network of projects, we use standard weighted graph G(V P , E P ), where weighted function is defined as w P : E P (G) → R, when w P (e) ≥ 0. We consider V P = P and we use Score for edges evaluation in the new graph G(V P , E P ).

Experiments

For the basic computation of the collaboration, we chose the terms "iphone", "wp7", "android" and apply it to the formula 3.

The results were limited to the collaborators with whose the person has collaborated together on the project at least once. We show centrality value of selected nodes in the Table 3. These centralities characterize the position of vertices in the network. We show in the Table 4 values of Attribute Score for person with nickname modder, in the Table 5 for person with nickname raja4567 and in the Table 6 for person with nickname raouf.

We can immediately notice that even though "modder" and "raouf" do not participate on many projects with "raja4567" (they have one common project), the AtributeScore is 0.01176471 and 0.01204819. For example "shankar00" participate on 2 projects with "raja4567" and the AtributeScore is 0.02469136. User "modder" has only one common project with "shankar00", but AtributeScore is strong, probably because "shankar00" not cooperate with many other persons.

Key Terms Computation for Developers

At first, we have calculated the keywords for the "modder", "raja4567" and "raouf". We have selected only the some terms for illustration (see Table 7). For comparison we marked some terms (bold text), which was used as a context between developers.

In the Figure 1 is whole network of collaborators for the selected terms "iphone", "wp7", "android". The edge weights are evaluated by Score. This subnetwork has 199 connected components (communities) with collaborating developers.

Second part of Figure 1 shows graph of the connected component which contain selected and highlighted developers. We can see that selected developers are not in the one community of Fig. 1. Synthetic collaborators network for the terms "iphone", "wp7", "android" and selected subnetwork with developers "modder", "raja4567" and "raouf" collaborators. They are connected, but the relation is too weak. They are not suitable for recommendation. We used our algorithm for spectral clustering [START_REF] Vojacek | Combined method for effective clustering based on parallel som and spectral clustering[END_REF] and we detect communities of more collaborated developers. Than we can recommend the "green" community, which contain developers with the stronger relation in the context of selected words.

Subnetwork of Projects

We chose the terms "iphone", "wp7", "android" and create subnetwork of projects. The graph of this subnetwork (see Figure 2) is not connected (contain 243 connected components) and most components are isolated vertices. When we extend the selected terms and create the new subnetwork in the context with terms "iphone", "wp7", "android" + "silverlight", than is obvious a importance of the term "silverlight" which connect more projects. Fig. 2. Extracted subnetwork of projects for selected terms "iphone", "wp7", "android" and the other subnetwork have terms "iphone", "wp7", "android" + "silverlight"

The Figure 1 and the Figure 2 were visualized using the program Gephi1 .

Conclusion

Research presented in this article is oriented to the strength extraction between persons based on their context in the CodePlex. The method was presented using the data collection from the CodePlex database, which contains information of the activities of developers in the project. The proposed method is usable for the development of collaboration network. The description of this network is based on the set of terms (as the input), which are used in the description of projects by the given developer. Using this method, we have obtained the new weight in the synthetic collaborators network. By means of the set of selected term, belonging to one (or more) persons, we can construct the subnetwork with only the context-related collaborators. This subnetwork can be very helpful in searching of the persons who are interested in the same area, defined by the selected term. It is usable for members of the project management, who need to find suitable developers specialized to certain area.

  CodePlex is Microsoft's open source project hosting web site. You can use CodePlex to find open source software or create new projects to share with the world. Codeplex.com has 11 years old, it is ranked 2,107 in the world, a low rank means that this website gets lots of visitors. Its primary traffic from United States and is ranked 3,175 in United States. It has 104 subdomains with traffic. It has 136,500 visitors per day, and has 436,800 pageviews per day. CodePlex is mainly used by developers for collaboration on projects, sharing source codes, communication and software development.

Table 1 .

 1 The CodePlex database tables

	Table	Number of lines
	User	96251
	Project	21184
	Discussions	397329
	RecentActivity	72285
	Membership	126759
	SourceCode	610917
	divided into different types: SourceCode, Discussion, RecentActivity and Membership
	(see Table 2).

Table 2 .

 2 The CodePlex activitiesActivity MeaningSourceCode records about added projects Discussion discussions about the project and the responses of individual users RecentActivity check-ins, task records, add Wiki information, notes about Release version etc Membership able to trace the users' participation in the projects and their assigned role

Table 3 .

 3 Centralities of developers

	User Degree Weighted degree Closeness Betweenness
	raja4567 34	7,69353	1,92	4948,5
	modder	4	0,0033	2,879	4143
	raouf	7	0,0366	2,879	0

Table 4 .

 4 Collaborators of "modder"

	Number Collaborators Projects Common projects Attribute Score
	1	modder	5	5	1
	2	doln	1	1	0,2
	3	draculus	1	1	0,2
	4	FreQi	1	1	0,2
	...
	13	shankar00	3	1	0,1666667
	...
	25	raja4567	81	1	0,01176471

Table 5 .

 5 Collaborators of "raja4567"

	Number Collaborators Projects Common projects Attribute Score
	1	raja4567	81	81	1
	2	senux	7	5	0,0602
	3	atechnikality	8	5	0,0595
	4	sagarjena	9	5	0,059
	...
	15	shankar00	2	2	0,025
	...
	408	raouf	3	1	0,012
	...
	429	modder	5	1	0,01176471

Table 6 .

 6 Collaborators of "raouf"

	Number Collaborators Projects Common projects Attribute Score
	1	raouf	3	3	1
	2	bvencel	1	1	0,33
	3	KathyWu	1	1	0,33
	4	srikanth602	1	1	0,33
	5	Lickie	1	1	0,33
	...
	15	raja4567	81	1	0,01204819

Table 7 .

 7 Key Terms for the persons "modder", "raja4567" and "raouf"

	number	t k	t k in T P modder	t k	t k in T P ra ja4567	t k	t k in T P raou f
	1	mediascout	1	licens	1	torchlight	1
	2	ne	0,7800623 distribut	0,7	resx	0,7017544
	3	sal	0,5980892 contributor 0,6934211	crunch 0,6028985
	4	movi	0,4556041	work	0,6413794	decenc 0,333333
	5	rapid	0,4191964	term	0,5994475	svt	0,3301587
	6	rockethub 0,401282	notic	0,5570145 blackberri 0,2729659
	7	myne	0,401282	modif	0,5359043	empti 0,2396313
	...
	14					android 0,1438451
	...
	143	wp7	0,1152854
	...
	1305			android 0,01322994
	...
	2572			iphon	0,00643989
	...
	2587			wp7	0,006402954

http://gephi.org/

Acknowledgment

This work was supported by SGS, VSB -Technical University of Ostrava, Czech Republic, under the grant No. SP2012/151 Large graph analysis and processing and by the Bio-Inspired Methods: research, development and knowledge transfer project, reg. no. CZ.1.07/2.3.00/20.0073 funded by Operational Programme Education for Competitiveness, co-financed by ESF and state budget of the Czech Republic.


Bartosz Marcinkowski
email: bartosz.marcinkowski@ug.edu.pl

Michal Kuciapski
email: michal.kuciapski@ug.edu.pl

A Business Process Modeling Notation Extension for Risk Handling

Keywords: Business Process Modeling Notation, Risk Management, BPMN Extension

During the years of prosperity, numerous organizations neglected numerous aspects of risk management. As systematic approach to handling identified risks is crucial to achieving success by the organization, modern business modeling standards and techniques are supposed to take risk-related features into account. The article is devoted to elaborating and exemplifying an extension aimed at risk handling for OMG's Business Process Modeling Notation (BPMN), one of the most prospective standards for business process modeling. After an introduction, key risk management concepts are discussed. Section 3 discusses extensions introduced within BPMN meta-model, while section 4 exemplifies proposed concepts. The article is concluded with a summary.

Introduction

Continuous and lasting for decades evolution of all-purpose and domain-centric notations resulted in creating modern modeling standards, such as Unified Modeling Language (UML), Business Process Modeling Notation (BPMN) or BPMS method, integrated in ADONIS business management solution. There is however plenty of room for further improvement. Needless to say, a few BPMN extensions were proposed by academic community and business modeling community.

Early versions of BPMNalthough often classified by practitioners as powerful on their own (comp. [START_REF] Harrison-Broninski | The Future of BPM. Part 2[END_REF])were enriched with business process goals as well as performance measures by [START_REF] Korherr | Extending the EPC and the BPMN with Business Process Goals and Performance Measures[END_REF]. [START_REF] Rodriguez | A BPMN Extension for the Modeling of Security Requirements in Business Processes[END_REF] presented a BPMN 1.0 meta-model with core element and extension that allowed incorporating security requirements into Business Process Diagrams aimed at increasing the scope of the expressive ability of business analysts. [START_REF] Magnani | BPDMN: A Conservative Extension of BPMN with Enhanced Data Representation Capabilities[END_REF] addressed relevant limitations of BPMN regarding weak data representation capabilities in comparison to competing standards, designed or adapted to business process modeling needssuch as ADONIS BPMS method or OMG's Unified Modeling Language profiles and custom approaches (comp. [START_REF] Przybylek | The Integration of Functional Decomposition with UML Notation in Business Process Modeling[END_REF]. Some of the concepts proposed were included later in BPMN 2.0 specification (Object Management Group BPMN, 2011), which was a significant revision that led the emerging standard the UML-alike multi-diagram way (comp. [START_REF] Wrycza | UML 2.0 in Information Systems Modeling[END_REF]. [START_REF] Stroppi | A BPMN 2.0 Extension to Define the Resource Perspective of Business Process Models[END_REF] offered solutions to strengthen the resource perspective of a business process model elaborated using BPMN 2.0 in order to improve the communication of resource structure, authorization and work structure between business analysts and technical developers. (Zor, Schumm and Leymann, 2011) enhance BPMN 2.0 modeling capabilities within manufacturing domain.

Business Process Modeling Notation extension capabilities are to be used by Object Management Group itself as a UML Profile for BPMN Processes Request For Proposal document (Object Management Group RFP, 2011) was issued. Having that said, the profile is not intended to develop notational capabilities but to provide a mapping between BPMN semantics and the profiled UML semantics as well as define XSLT transforms between the UML XMI for the profile and the BPMN 2 XSD and QVT transforms between the UML and BPMN 2 meta-models.

During the years of prosperity, numerous organizations neglected numerous aspects of risk management. As systematic approach to handling identified risks is crucial to achieving success by the organization, modern business modeling standards and techniques are supposed to take risk-related features into account. The article is devoted to elaborating an extension aimed at risk handling for Business Process Modeling Notation, one of the most prospective standards for process modeling.

Basic concepts of risk management

According to (ISACA, 2006), risk management is the process of identifying vulnerabilities and threats to the information resources used by an organization in achieving business objectives, and deciding what countermeasures, if any, to take in reducing risk to an acceptable level, based on the value of the information resource to the organization. Risk itself is defined as the combination of the probability of an event and its consequence (International Organization for Standardization, 2009). Risks may be divided into (International Business Machines Corporation, 2007):

 business-driven risks, strategic in nature and aimed at protecting the business and keeping it accessible whenever and from whoever in support of continuous business operations as well as compliance with industry and government regulations;  data-driven risks, dealing with the availability of data and information in all of its different forms as used by the organization, including paper-based data;  event-driven risks, focusing on actual events that create risk to business continuity and viability, such as natural disasters, thefts and IT attacks.

It is risk mitigation procedure that is one of the relevant aspects of risk management to be included in the elaborated extension. Based on the canon of literature in the field of risk management, four standard ways to handle risk are commonly acknowledged (comp. DeLoach and Temple, 2000):

 reduce,  retain,  avoid,  transfer. [START_REF] Husdal | The Six Ways of Dealing with Risk[END_REF] proposes to set up a wider framework of risk management by discussing exploit and ignore strategies and adding it to the list.

Risk handling in BPMNstandard features and extensions

Distinctive features of BPMN include, in particular, very extensive semantics of events. In addition to the division of events into start, intermediate and end ones, the notation development team proposed twelve types of events, distinguished events that are thrown and caught as well as proposed the possibility of interrupting or continuing the flow of source activity at the time of event occurrence. It is the error event that is particularly important type of event from the perspective of risk management. Such event type points to an exception in the underlying activity. The functionality of the described event may in fact be used to assign identified risks to processes, subprocesses or activities to which these risks apply. Indication of potential points of error and how to design their handling is therefore the starting point of expansion of the standard BPMN business process diagram. BPMN standard, however, does not support formal specification of the identified risks. A business analyst can attempt fulfilling that task by introducing text annotations, which are among the build-in BPMN artifacts, but it is a technique of low clarity and precision. Therefore, the standard was extended with the modeling category of risk factor, characterizing a potential risk in terms of the type, likelihood and impact on business process as a whole. Analogously as in the related publications [START_REF] Kuciapski | Risk-oriented Modeling in Business Process Specification[END_REF], [START_REF] Kuciapski | Risk Management in e-Learning Projects of Courses Development and Implementation[END_REF] to both the likelihood and impact ranks from 1 to 5 range were assigned, with a value of 1 indicates a low occurrence probability (impact), while the value of 5very high occurrence probability (impact).

Risk factor is designed as an independent modeling category due to its complex nature. From the perspective of BPMN meta-model, it was reasonable to assign it to a group of artifacts. Therefore, the RiskFactor is implemented as a child element of Artifact, and supplemented with the additional properties -occurenceProbability and impact (see Fig. 1). A single RiskFactor can be associated with multiple types of risks, but the specification of at least one RiskType is mandatory. Types of risk are classified as integral parts of risk factors, as highlighted with the use of the composition. RiskType is treated in terms of an abstract modeling category. Proposed extension of the standard recognizes five types of risks, but this list may be expanded according to the needs of the end user. Each proposed type of risk is assigned an individual notation. PhysicalResourceRisks can be related to a list of Resources, as HumanResourceRiska list of Participants.

Risk factors are assigned to BPMN sequence flows. For this purpose, a standard modeling category of association or placement directly by the relevant sequence flow is used. It should be emphasized that with version 2.0 of the BPMN notation, an Association is distinguished from a DataAssociation, hence the association in the current version of the standard is not document-oriented. A single identifiable RiskFactor can be attributed to multiple SequenceFlows. Naturally, from the standpoint of a sequence flow, binding risk factors is optional. The extension also includes the possibility of decomposing risk factors onto component risk factors.

Fig. 1. Risk factors and BPMN meta-model

Each identified risk ought to be assigned with a solution. In the proposed extension, tasks dedicated to handle risk factors are distinguished from tasks forming a natural flow of process or sub-process. In order to achieve that, BPMN meta-model is supplemented with additional kind of Task, i.e. RiskHandler, along with accompanying markup notation (Fig. 2). In order to provide basic compatibility with ADONIS risk analysis process extension for proprietary BPMS notation, both RiskHandlers and RiskFactors introduce icons elaborated within mentioned solution. Since a Task is a special case of an Activity, RiskHandler inherits the characteristics of activities. In the context of risk management the significance of Resources assigned to Activities -Contractors in particularshould be emphasized.

It is the mitigationMethod that is an integral property of a RiskHandler. Based on the list of strategies included in section 2 of the current article, six ways to handle risk are proposed: Reduce, Retain, Avoid, Transfer, Exploit and Ignore. As the method is expressed as a String, an enumeration called RiskMitigationMethod is designed for the purpose of storing potential values of the property.

Fig. 2. Risk handler as a child element of BPMN Task

BPMN Extension for Risk Handling exemplified

The proposed extension was tested within business process modeling project, involving specification of diverse business processes for real estate developer companies. It is a sub-process that illustrates the procedure for managing architectural contests that is the business functionality selected for the current paper (Fig. 3). Analysis of the initial BPMN diagram reveals that it is legitimate to consider a number of risks that the company, responsible for the investment, should foresee in the context of conducting the architectural contest. Thus, risks itemized by the company included the risk of insufficient number of entries to the contest, receiving entries that are produced using non-standard toolkits (and thus containing files in nonsupported graphics formats), or occurrence of legal uncertainties as to the authorship of the project. These risks are addressed by risk factors, respectively, Insufficient number of entries, Unsupported file format and Legal issues raised. It should be noted that within the proposed solution of the risk management issues, each identified risk factor is to be attributed to a separate event. Thus, on the border of an activity a lot of intermediate events, responsible for initiating various factors, may be placed.

Two former risk factors are classified as PhysicalResourceRisks, while the latter is classified as a HumanResourceRisk. Risk factor Insufficient number of entries is rather unlikely to occur (rank 2), but has a great impact on the owning sub-process (5). It is Publish information regarding inconclusive outcome that is the task devoted to handling the risk factor. On the other hand, receiving Unsupported file format is more common (rank 3) while having virtually no impact on the sub-process at all (rank 1). Should the risk factor occur, risk handler Extend format conversion request is invoked. Risk factor Raising legal issues is very rare (rank 1) but has significant severity (rank 4). A task Verify entry's sufficiency in law was designed to handle risk.

Summary

It was elaborating an extension aimed at risk handling for Business Process Modeling Notation that was the goal of the current article. As BPMN functionality for risk modeling is very limited, even basic risk-oriented framework required introducing custom modeling categories, i.e. RiskFactors, RiskTypes, RiskHandlers as well as RiskMitigationMethods. Owing to the consistent practice of publishing metamodels for standards maintained by Object Management Group, the categories were seamlessly integrated with BPMN meta-model and designed so that the subsequent expansion was possible. Practical applications of proposed extension were exemplified by illustrating the procedure for managing architectural contests.

Fig. 3 .

 3 Fig. 3. Procedure for managing architectural contests developed using BPMN Extension for Risk Handling


Agnieszka Jastrzebska
email: a.jastrzebska@mini.pw.edu.pl

Wladyslaw Homenda

Modeling Consumer Decision Making Process with Triangular Norms

Consumer decision making processes are conditioned by various forces. Recognized premises are being constantly reevaluated and future decisions are made in connection with previous ones. Therefore, authors propose an approach to decision making modeling based on pairs of vectors describing attitudes towards certain attributes influencing consumer's decision. As a result, possible is a description of preferences based on multiple evaluations of one feature. Methodological approach allows to reevaluate opinions, which even though were taken in past, but still influence current decisions. In the article we discuss how triangular norms can be applied to decision making modeling based on information encouraging the choice gathered in paired vectors. Developed methodology is based on combining different known triangular norms for given pairs of vectors representing various consumers, who are facing the same decision. The choice of preferences evaluation was performed so that they would represent different real-life situations. Results of applied processing based on combinations of triangular norms are compared. Drawn are conclusions about various processing properties of aggregation operators. Suggested is, in which cases, certain triangular norms describe real-life processes accurately. Shown and described are also examples of operators, which are not suitable at all.

Introduction

Decision making processes are driven by various needs. Needs determine our actions, direct us towards reducing motivational tension. As we described in [START_REF] Homenda | A Jastrzebska, Modelling consumer needs[END_REF], proposed were several hierarchies of how humans proceed with satisfying their needs. A very interesting perspective on the grounds of decision making processes was presented by K. Lewin,[START_REF] Lewin | Field theory in social science[END_REF]. This article continues the discussion started in [START_REF] Homenda | A Jastrzebska, Modeling Consumer's Choice Theory: Using Fuzzy Sets and their Generalizations[END_REF] of how the theory of psychophysical field can be applied to consumer decision making modeling using fuzzy sets and their generalizations.

First, we would like to recall basic notions of how a decision making process can be perceived as a process inside the psychophysical field and how multivalued logic operators can be applied to model it. In section 3 we discuss various triangular norms, which are applied in section 4 to modeling. Presented is a case study of one particular choice. Decisions are based on fuzzy information, which in fact is very common. Developed methodological approach allows us to reevaluate consumer's opinions, which even though had been taken in past, but still influence current decisions. We compare aggregation possibilities incorporated in different triangular norms and suggest, which cases describe real-life processes most accurately.

Preliminaries

Psychophysical field is an abstract term, describing decision making process. It is formed as a combination of three factors:

1. The field (which can be perceived as a set of conditions). Field contains all motivational stimuli arising when subject thinks about the decision. Conditions of the decision are bounded by available knowledge and previous experiences. The field includes not only premises speaking for or against given choice, but also all conscious and subconscious factors, which may influence the decision. 2. Processes, which describe a way of how one makes decisions. It is directly connected with individual's behavioral patterns. Processes are conditioned by customer's cognitive abilities, rationality and susceptibility to behavioral biases. 3. The decision -called also the gestalt -it is the outcome of the decision making process.

In this article we would like to focus on how triangular norms can be applied to decision making modeling. We would restrict our discussion to positive premises only, though our future research will include bipolar information aggregation techniques. Due to this restriction, we would be able to model only some fraction of real-life processes, but presented techniques can be also adapted for cases based on both positive and negative premises.

The field (all conditioning factors) will be grouped into two sets: premises and priorities. These are all recognized motivational stimuli, which ground our decisions. Processes are represented by triangular norms, which we apply. The decision (gestalt) is the output of our computations.

As introduced nomenclature may raise some questions, at first, we would like to explain used notions. While speaking about decision making process, we would describe perceived stimuli in the terms of premises and priorities. A premise, in our understanding, is received and decoded information influencing given decision. We assume that one can evaluate the importance of a premise as a number from the range [0, 1], where the greater is the value of the premise, the stronger is positive influence of the premise on the decision. This allows us to apply a flexible scale of evaluation. For each customer we discuss the same set of premises (the decision is made about the same product), but with different evaluations.

While computing the result of the decision making process we introduce aggregating operators, which allow to obtain a decision as a number from the [0, 1] range. The greater is the output of our computations, the more convinced is given customer about the decision. What did we gain? First of all, discussed models describe real life situations better. We are able to base and compute imprecise information. Even though we may calculate the decision in the crisp form (binary response), we are intentionally highlighting that plenty of real-life decisions are perceived rather as weak or strong attitudes.

Premises describe customer's motivation towards certain products or services. In terms of needs theory (see [START_REF] Homenda | A Jastrzebska, Modelling consumer needs[END_REF]) premises are motivational stimuli, which elicit, control and sustain certain behaviors. These are all factors, which arise when an individual thinks of given decision in more general terms. It can be somehow called an initial or an a priori motivation. Authors treat all premises as a set of infinite amount of opinions or attitudes, from which while discussing a particular decision accounted and considered is only a relatively small and countable subset. Why small? Because people tend to simplify rather than complicate reality. Moreover, facing time constraints, people are aware that in order to efficiently manage one's time, even though there might be million possibilities, only a few are really worth discussing.

Second term present while discussing decision making processes are priorities. Authors intend to use this term in the context of a second set of beliefs (or in other words as a second set of motivational stimuli). Priorities allow us to take into account reassessed attitude towards one particular choice. In this paper, term premises concerns more general case, while priories describe certain precise choices. Analogically, priorities can be expressed scaled, for example as a number from the interval [0; 1]. Priorities are recognized and evaluated later than premises, and their purpose is to provide a perspective of how one particular choice satisfies one's needs. In this context, they may be perceived as an aposteriori motivation, arising when subject has faced particular product or service. Of course, a set of priorities evaluations might be drastically different than premises.

Triangular norms

In this section we recall basic notions of fuzzy sets and generalization of fuzzy connectives maximum and minimum to triangular norms. We will be expressing fuzzy sets in the form of membership functions. Namely, a fuzzy set A defined in the universe X is a mapping µ : X → [0, 1] or µ A,X : X → [0, 1] if the names of the set and the universe should be explicitly stated.

The Zadeh's model of fuzzy sets can be described as a system similar to set theory (F (X), ∪, ∩, ), where F (X) denotes fuzzy sets over the universe X and ∪, ∩, denote union, intersection and complement. This system is clearly interpreted as ([0, 1] X , max, min, 1-), where [0, 1] X denotes all mappings from the universe X into the unit interval [0, 1], i.e. the space of membership functions, and max, min and 1-applied to membership functions implement union, intersection and complement. We do not pay attention to the interpretation of fuzzy sets in terms of a lattice L X .

In this paper study of fuzzy sets system was enriched with triangular norms used in place of the max and min operators. Note: max and min are triangular norms as well.

Triangular norms, i.e. t-norms and t-conorms, are mappings from the unit square [0, 1] × [0, 1] onto the unit interval [0, 1] satisfying axioms of associativity, commutativity, monotonicity and boundary conditions (cf. [START_REF] Klement | Triangular norms[END_REF][START_REF] Schweizer | Probabilistic Metric Spaces[END_REF] for details), i.e.: Definition 1. t-norms and t-conorms are mappings p :

[0, 1] × [0, 1] → [0, 1],
where p stands for both t-norm and t-conorm, satisfying the following axioms:

1. p(a, p(b, c)

) = p(p(a, b), c) associativity 2. p(a, b) = p(b, a) commutativity 3. p(a, b) ≤ p(c, d) if a ≤ c and b ≤ d monotonicity 4. t(1, a) = a for a ∈ [0, 1] boundary condition for t-norm s(0, a) = a for a ∈ [0, 1]
boundary condition for t-conorm t-norms and t-conorms are dual operations in the sense that for any given t-norm t, we have a dual t-conorm s defined by the De Morgan formula

s(a, b) = 1 -t(1 -a, 1 -b)
and vice-versa, for any given t-conorm s, we have a dual t-norm t defined by the De Morgan formula

t(a, b) = 1 -s(1 -a, 1 -b)
Duality of triangular norms causes duality of their properties. Note that the min/max is a pair of dual t-norm and t-conorm. The selected known t-norms and their dual t-conorms are given in Table 1. Note that Hamacher product is

product x • y probabilistic sum x + y -x • y Lukasiewicz max(0, x + y -1) bounded sum min(a + b, 1) nilpotent minimum    min(x, y) if x + y > 1 0 otherwise nilpotent maximum    max(x, y) if x + y < 1 1 otherwise drastic        y if x = 1 x if y = 1 0 otherwise drastic        y if x = 0 x if y = 0 1 otherwise Hamacher product    0 if x = 0 = y x • y x + y -x • y otherwise Einstein sum x + y 1 + x • y
the chosen representative of the parametric class of Hamacher norms. t-norms and t-conorms are bounded by minimum t-norm and maximum t-conorm, i.e. for any t-norm t, any t-conorm s and any x, y ∈ [0, 1]:

t(x, y) ≤ min(x, y) ≤ max(x, y) ≤ s(x, y) (1)
We will be discussing consumers' decision making process modeled with triangular norms. In this study consumers' decision making is interpreted in the following way:

• first, a consumer attempts purchasing a product or a service considering general needs for it. This solicitude results in a series of necessity factors corresponding to needs. The necessity factors are expressed as values of a membership functions and we call them premises • then, (s)he confronts the general needs with properties of a concrete item or offer, which fits the general need to some extent. The confrontation produces fitting factors, again expressed as values of a membership function. We call the fitting factors priorities, • we assume that the property of a given item/service cannot increase corresponding need, it rather may soften the need. We say that priorities moderate premisses and implement the moderation with applying a t-norm. Note that according to formula 1 the result of moderation cannot exceed weaker of the premise and the priority, • finally we aggregate moderated premisses and priorities with a t-conorm. We assume here that needs vote for purchasing, therefore they either strengthen each other, or at least not weaken themselves. Note that the formula 1 shows that the aggregation with any t-conorm will produce a result not weaker than the stronger moderated premise/priority. In this study we consider only positive premisses. A discussion on negative premisses, i.e. premisses which vote against purchasing, is not in the scope of this paper. We only wish to note that initial discussion on this subject was undertaken in [START_REF] Homenda | A Jastrzebska, Modeling Consumer's Choice Theory: Using Fuzzy Sets and their Generalizations[END_REF]. That attempt, utilizing balanced fuzzy sets [START_REF] Homenda | Balanced Fuzzy Sets[END_REF] in decision making, combines both types of premisses: positive and negative. In frames of another strive positive and negative premisses are aggregating separately and then final decision is made on these aggregation. So called intuitionistic fuzzy sets [START_REF] Atanassov | Intuitionistic fuzzy sets[END_REF][START_REF] Atanassov | More on intuitionistic fuzzy sets[END_REF] or vague fuzzy sets [START_REF] Gau | Vague sets[END_REF] can be used as vehicles for such aggregation.

Case Study

To be able to compare described in section 3 models and observe how their properties are reflected while modeling decision making processes, we introduce a case study. In this paper we will continue a discussion on a decision making process regarding purchase of a car (by analogy to: [START_REF] Homenda | A Jastrzebska, Modeling Consumer's Choice Theory: Using Fuzzy Sets and their Generalizations[END_REF]). In order to be able to compare results, we will discuss the same set of eight pairs of premises and priorities for five different customers.

We will be analyzing following attributes encouraging purchase of a car: • if you have to take care of babies, it is easier to transport them in a car than by public transport, • shopping with a car is very convenient, • in a city, where decision maker lives, car allows you to go faster than by bus or by tram,

• having a nice car manifests consumer's good taste and his wealth,

• with a car you can easily make weekend trips to nearby places,

• car allows you to travel at any time you'd like, you are not dependant on any timetables, • if the weather is bad, driving a car is better than waiting on the bus stop,

• car allows to transport plenty of luggage without overworking.

First, we describe customers and their vectors of premises and priorities. Next, in subsection 4.2 we discuss methodology of applying triangular norms for computing the decision.

Consumer's vectors description

The analysis revolves around five customers (named A, B, C, D and E). All of them are discussing the same set of attributes regarding a purchase of a car. Selection of vectors of premises and priorities describing customer's preferences was performed by authors. We chose these particular values in order to capture different real-life situations.

Customer A was assigned with following vector of premises:

A premises = [ 0.00, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20 ]

Customer A evaluated all premises as weak ones. First premise is considered as not important at all (is equal to 0). He is not convinced that having a car is necessary. Similarly, vector of priorities for customer A expresses his lack of strong positive opinions regarding one particular car.

A priorities = [ 0.00, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20 ]

A is consequently convinced, that there is no strong motivation for him to buy a car. We expect that consumer his final decision regarding the purchase of a car will be weak.

Customer B was assigned with following vector of premises:

B premises = [ 1.00, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20 ] (4)
B has evaluated first premise as extremely important. He has small babies and having a car would be very helpful to transport them. The rest of premises were evaluated as rather not influencing. Below shown is vector B priorities , which gathers customer B's priories evaluation towards one particular car.

B priorities = [ 1.00, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20 ]

Analogically, first priority was evaluated as extremely important -he believes that this particular car is suitable and would play its role as a help while transporting his family. The rest of priorities still have weak impact on his behavior.

Customer C is described by a following vector of premises:

C premises = [ 0.80, 0.80, 0.80, 0.80, 0.80, 0.80, 0.80, 0.80 ] (

All premises were recognized as very important. Analogically, C's vector of priorities C priorities contains high values.

C priorities = [ 0.80, 0.80, 0.80, 0.80, 0.80, 0.80, 0.80, 0.80 ]

C is strongly convinced that buying car is necessary. Moreover, this one particular car he is analyzing meets his expectations. We expect that C's decision should be strong positive.

Customer D has following values attached to premises regarding purchase of a car in general:

D premises = [ 0.80, 0.30, 0.70, 0.00, 0.10, 0.60, 0.90, 0.30 ]

His preferences are varied. There are several premises recognized as important and several recognized as very weak. There is even one attribute (premise number 4: car as a way to manifest one's wealth and social status), which in D's opinion initially does not matter at all. Vector D priorities describes D's priorities towards particular car.

D priorities = [ 0.10, 0.80, 0.20, 0.90, 0.60, 0.10, 0.00, 0.80 ]

This customer's strengths of all priorities are drastically different than strengths of corresponding premises. When D has faced this one particular decision, he drastically reevaluated his opinions. We will observe how different moderating operators would cope with these vectors. Situation captured in D premises and D priorities corresponds to a case, when facing one particular choice we recognize several new important features of given product. In such case, old premises lose their impart on the final decision. This can be caused for example by very persuasive marketing communications, when customer starts to analyze new priorities, which concern one particular car. Successful marketing campaign makes customer D believe that these new projected features of advertised car are even more important than original premises.

Customer E was assigned with following vector of premises: E premises = [ 0.80, 0.80, 0.80, 0.80, 0.80, 0.80, 0.80, 0.80 ] (

Customer E has reviewed his opinions regarding purchase of a car in general and decided that all premises are very strong. But when one particular car was analyzed, he reevaluated the importance of named priorities and it turned out that all attributes are rather insignificant. Vector E priorities presents reevaluated priorities.

E priorities = [ 0.20, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20 ] (11)

Situation described with vectors E premises and E priorities happens when at first we are strongly convinced that some product is a necessity. Next, when we face a particular decision, we are surprised to notice that what has been so appealing has lost its charm and we are rather disappointed.

Results of aggregation

In following subsections we discuss results obtained for A, B, C, D and E after applying operators described in section 3. We discuss three different methodologies of moderation and aggregation:

1. various t-norms for premises and priorities moderation and max t-conorm for final decision calculation, 2. min t-norm for premises and priorities moderation and various t-conorms for final decision aggregation, 3. various t-norms for premises and priorities moderation and their dual tconorms for final decision calculation.

Chosen strategy of applying triangular norms was dictated by literature review and our intuition. Authors picked popular triangular norms and combined them in order to obtain different results.

Moderation with different t-norms and aggregating with max t-conorm

In this subsection we will discuss decisions made for customers A, B, C, D and E basing on maximum t-conorm and vectors of premises and priorities moderated with different t-norms. At first, we would like to show exemplar outputs of moderation. Please note that all calculations were performed according to formulas given in Table 1.

Table 12 In this first case, consumers A, D and E are not convinced about the purchase of given car. Even though some of them have recognized several strong priorities (or premises) the process of moderation disregarded them and the decision was computed as a weak one.

Second column contains values received by maximum t-conorm aggregation on vector of moderated premises and priorities. Moderation was performed using product t-norm. In the second case, only B and C made strong positive decisions. Through properties of multiplication of numbers not greater than 1 the output of moderation contains mostly low numbers. Therefore, max t-conorm as an aggregation operator is more than necessary so that we can somehow balance low results received through multiplication. Applied product t-norm and maximum t-conorm for E gave us result equal to 0.16. Previously discussed combination of operators also returned equally weak choice. The weakest value of aggregated decision was computed for A. Product t-norm strongly reflected the fact that all premises and priorities for A were weak ones.

Third column of table 13 contains decisions obtained with a combination of Lukasiewicz t-norm and maximum t-conorm. Analogically to the previous case, consumers B and C are the only ones, who are convinced about the purchase of a car. Consumers A and E's decisions are equal to 0 -they are absolutely indifferent towards the decision regarding a particular car. This is first combination of tnorms and t-conorms, which returns such a strict output.

Fourth column in table 13 presents values obtained after applying Nilpotent minimum t-norm and max t-conorm for A, B, C, D, and E. Results of aggregation obtained with Nilpotent minimum t-norm and maximum t-conorm are close to results obtained using Lukasiewicz and maximum norms. B and C are strongly convinced about the purchase of the car, while A, D and E would not buy the car. In this case output for C is more satisfying (is higher) than in the previous case -enhanced was the fact that C's both vectors have all strong positive values.

Fifth column contains the results obtained after applying Drastic t-norm and maximum t-conorm. As we expected, all uncertain cases were eliminated. Positive decision was computed only for consumer B.

Sixth column in table 13 presents the output of moderation and aggregation using Hamacher product t-norm and maximum t-conorm for all customers. Noticeable is that B and C are strongly convinced about the purchase of the car. At the same time A, D and E's decisions are weak positive.

Moderating with min t-norm and aggregating with various t-conorms

Second part of our experiment was to observe the differences between decisions aggregated using different t-conorms. In this approach in each case premises and priorities were moderated using minimum t-norm. Results obtained using min t-norm for moderation and various t-conorms for output calculation are unsatisfactory. Apart from the first and fourth case, when we applied min-max dual operators and minimum/Nilpotent maximum norms, all other results compute strong positive decisions for each customer. Especially undesirable effects obtained were for bounded sum and Drastic t-conorms.

We find second methodology rather unsuitable. Nevertheless, basing on discussed results, we observed how different t-conorms aggregate sequences of the same values. For example, let's discuss aggregation of an 8-element sequence of 0.2s. We observe that two t-conorms, namely maximum and Nilpotent maximum maintain the same result and it is 0.2. All other discussed t-conorms gradually saturate and tend to 1. Norm, which is returning 1 instantly is Drastic t-conorm. Norm, which is tending most slowly to 1 is bounded sum. The higher are the values of aggregated sequence, the faster the result tends to 1. For a sequence of 0.8s, only after three operations the result is saturated. Of course, by definition of triangular norm, the result of aggregation is always bounded by 1.

In consequence, gradual saturation of the result means that certain triangular norms, including bounded sum, probabilistic sum and Einstein sum incorporate following property: the more positive motivational stimuli one recognizes the closer the decision gets to 1. Second corollary is following: one strong positive argument (moderated premise with priority) can be replaced by several weak arguments. Discovering these properties is very important from the point of view of consumer decision making modeling. Presented properties of aggregation operators are highly desirable -economists prove that people tend to simplify cognitive processes. What does it mean in the context of decision making? Saturation of named t-conorms allows us to model a situation, when customer, even though there might be an infinite number of premises, is able to efficiently make a decision basing on relatively small set of arguments. Presented t-conorms allow to represent nontrivial aspects of the decision making processes, including behavioral biases. We believe that applying them in the neoclassical theory of consumer's choice might be increase its accuracy and would allow us to describe real-life situations more accurately.

Moderating with different t-norms and aggregating with their dual t-conorms Last, but not least, we would like to discuss the decisions regarding purchase of a car computed basing on various dual t-norms and t-conorms.

Obtained results prove again that the choice, of which operators apply for computing the decision has major impact on the output. The differences between particular results are substantial. It is vividly seen in the case of customer C, whose decisions vary from 0 to 1, depending on the applied norm. Variety of properties incorporated in different aggregating operators would allow us to use ones, which would describe decision making phenomena the best. Decisions obtained for D and E are analogical to ones received with the first methodology. We find outputs computed using operators product/probabilistic sum and Hamacher product/Einstein sum for D and E as too high. These two vectors were constructed to reflect consumers who are undecided (D) or who are disappointed with one particular car described by vector of priorities (E).

Customers B and C were assigned with strong positive decisions in all cases, except from one (Drastic t-norm and t-conorm for consumer C). Comparing the third methodology with the first and the second one, we see that C's decisions were computed before as weaker. For further research left is the topic, which methodology is more suitable.

Finally, we'd like to discuss decisions computed using dual t-norms and tconorms for consumer A. Third approach computed rather optimistic results. In comparison, for A's decisions, first methodology in all cases computed values non greater than using the third approach. Third approach for the following three dual operators: Lukasiewicz t-norm/bounded sum, Nilpotent minimum/Nilpotent maximum, Drastic t-norm and Darstic t-conorm computed A's decisions as zeros. We find this results as too conservative, since these norms disregard all preferences, which even though are weak, but still they exist. As we mentioned in [START_REF] Homenda | A Jastrzebska, Modeling Consumer's Choice Theory: Using Fuzzy Sets and their Generalizations[END_REF], Kahneman and Tversky proved that people tend to overweight small probabilities and underweight moderate and high probabilities, c.f. [START_REF] Wakker | Prospect Theory: For Risk and Ambiguity[END_REF]. Therefore justified would be the choice of conservative operators for the case of consumer C and rather optimistic operators for A.

Conclusions

In the article we discuss how different triangular norms can be applied to model decision making processes based on positive premises only. We support our paper with the case study of five customers who decide about one particular car basing on the same set of attributes. These five cases represent different reallife situations of people, who show varied attitudes towards both a decision regarding purchase of a car in general and different opinions regarding one particular car, about which the decision is made. We apply three methodological approaches of how to calculate the decision using t-norms and t-conorms. First one uses various t-norms for premises and priorities moderation and maximum t-conorm to compute the output. Second approach applies minimum t-norm for vector's moderation and various t-conorms for decision calculation. Third approach uses dual t-norms and t-conorms. First and third methodologies bring us satisfactory results. Second one we find rather not suitable. We noticed that several t-norm/t-conorm combinations would allow us to model more conservative (weak) decisions, while some other are optimistically enhancing the result. An example of a set of operators strengthening the decision is Hamacher product and Einstein sum. Examples of t-norms, which applied to vectors of premises and priorities compute rather weak decisions, for almost each t-conorm, are Lukasiewicz, Drastic and Nilpotent minimum t-norms. Important conclusions were introduced while discussing aggregation of sequences using bounded sum, probabilistic sum and Einstein sum norms. Saturation of decision computed using named operators ideally reflects human's tendency for simplification. Presented triangular norms allow to represent nontrivial aspects of the theory of consumer's choice. Possibility of incorporating behavioral biases into consumer decision making models would allow us to develop a methodology, which would describe real-life phenomena more precisely and more extensively.

Table 1 .

 1 Selected

triangular norms, dual t-norms and t-conorms are placed in one row t-norm t-conorm minimum min(x, y) maximum max(x, y)

  contains exemplar results of premises and priorities moderation using minimum t-norm for consumers A and C. Next, we would like to discuss final decisions obtained for consumers A, B, C, D and E after applied maximum t-conorm. Table13contains values of decisions aggregated with max for vectors moderated using various t-norms. The top row of table 13, contains information about applied t-norms and t-conorms.Results of minimum t-norm and maximum t-conorm (first column) show that only B and C are strongly convinced about the purchase of one particular car. All other customers express weak opinions. Consumer B represents a situation, when gathered is one strong positive information and several weak premises and priorities. Decision computed for consumer C is also strong positive one, what is no surprise, as all evaluations both of premises and priorities are high.

		A: C:	0.0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8	(12)
		minimum product Lukasiew. Nilpotent Drastic Hamacher
		maximum maximum maximum maximum maximum maximum
	A:	0.20	0.04	0.00	0.00	0.00	0.29
	B:	0.90	0.90	0.90	0.90	0.90	0.90	(13)
	C:	0.80	0.64	0.60	0.80	0.00	0.67
	D:	0.30	0.24	0.10	0.30	0.00	0.28
	E:	0.20	0.16	0.00	0.00	0.00	0.19

In the same fashion calculated were moderated values for consumers B, D and E. Analogically, we moderated premises with priorities using other t-norms, namely: product, Lukasiewicz t-norm, Nilpotent minimum, Drastic t-norm and Hamacher product. Due to space limitations these calculations are not discussed to greater extent. Please note, that none of results obtained via moderation using t-norms different than minimum would give us values greater than ones received with the minimum (property explicitly stated in formula 1).

  Table 14 compares final decisions obtained using t-conorms as aggregation operators. The top row of table 13, contains information about applied t-norms and t-conorms.

		minimum minimum minimum minimum minimum minimum
		maximum prob. sum bound. sum Nilpotent Drastic Einstein
	A:	0.20	0.79	1.00	0.20	1.00	0.89
	B:	0.90	0.98	1.00	1.00	1.00	0.99	(14)
	C:	0.80	1.00	1.00	1.00	1.00	1.00
	D:	0.30	0.71	1.00	0.30	1.00	0.81
	E:	0.20	0.83	1.00	0.20	1.00	0.92

  Table which t-norm and t-conorm was applied in order to compute particular decision.

		minimum product Lukasiewicz Nilpotent Drastic Hamacher
		maximum prob. sum bounded sum Nilpotent Drastic Einstein
	A:	0.20	0.23	0.00	0.00	0.00	0.58
	B:	0.90	0.92	0.90	0.90	0.90	0.98
	C:	0.80	1.00	1.00	1.00	0.00	1.00
	D:	0.30	0.60	0.20	0.20	0.00	0.78
	E:	0.20	0.75	0.00	0.00	0.00	0.91

Acknowledgment

The research is partially supported by the National Science Center, grant No 2011/01/B/ST6/06478.