1 Introduction

Nowadays, knowledge has the powerful influence which leads business organization to survive in the competition among firms. Knowledge sharing stands for exchanging and transmitting knowledge among individuals, groups, and organizations for the purpose of improving organizational competitiveness [1]. Knowledge sharing is a fundamental means through which employees can contribute to knowledge creating, innovation, and ultimately the competitive advantage of the organization [2]. Many researches have shown that knowledge sharing and fusion are positively related to reductions in production costs, faster completion of new product development projects, team performance, firm innovation capabilities, and firm performance [3]. Due to the advantages of knowledge sharing, many organizations pay attention to knowledge management system to support knowledge sharing.

Collaboration is a promised method which provides benefits of knowledge sharing to us. Collaboration is a process in which two or more members from different area participate in knowledge transmitting process for achieving a common task or a goal [4]. During collaboration process, synergetic effects among team members would contribute to generate novel knowledge. So, collaborations which derive powerful synergetic effects are required in teamwork. However, the way to produce effective collaboration is implicit because collaboration is a complex, multi-dimensional process which is characterized by constructs such as coordination, communication, meaning, relationships and trust [5]. Thus, this study aims to propose a mathematical model of collaboration mechanism in order to investigate the process.

Channel Theory [6] is utilized to achieve this goal, and Chu space [7] is adopted to account for knowledge in collaboration system. Chu space is a mathematical construction, which represents scheme of infomorphism. To verify the proposed model, an example of collaboration in an international design workshop is shown.

2 Literature Reviews

2.1 Related Works

There are many researches have found that a team which consists of different disciplines members succeeded to create new idea [8, 9]. According to these researches, the combination of diverse backgrounds members played an important role to bring about the new idea. As these examples show, different perspective is an important factor to create effective collaboration.

During collaboration process, synergetic effects among team members contribute to generate novel knowledge. However, there are a few researches which focus on qualitative analysis in collaboration process. Thus, a representation model of collaboration mechanism is proposed in order to support the qualitative analysis. The model is built based on Channel Theory. Channel Theory provides a logical framework to discuss transition of meaning through a collaboration.

Channel theory has been used in various fields. For example, Suto et al. have proposed a representation model for communication medium with Channel Theory [10]. This model is used to describe semantic information flow, which is corresponding to a kind of medium. Kawakami et al. have proposed a framework of modeling that involves diversity and context dependencies base on Channel Theory [11]. It has the potential to describe diverse understanding based on the information flows. Schorlemmer [12] proposed a formalization of knowledge sharing scenarios by using diagram in the Chu category. Basic ideas of Channel Theory and Chu spaces are referred briefly in the following sections.

2.2 Channel Theory

Channel Theory provides a mathematical framework of qualitative theory of information. The basic concepts of Channel Theory consist of classification, local logic, infomorphism, and information channel.

A classification: \({\langle A=tok(A),typ(A),\mathrm {\models }_A\rangle }\) consists of a set of objects to be classified; tok(A), called the “tokens of \( {A}\),” a set of objects used to classify the tokens; \( typ(A)\), called the “types of \( A,\)” and a binary relation \(\mathrm {\models }_A\) between \( tok(A)\) and \( typ(A)\) indicating the types into which tokens are classified.

Given a classification \( A\), a pair \(\langle {\varGamma },\bigtriangleup \rangle \) of subsets of \( typ(A)\) is called a “sequent of \( {A}\).” A token \( a \in ~tok(A)\) \(\mathbf {satisfies}\) \(\langle {\varGamma },\bigtriangleup \rangle \) if a is of type \(\alpha \) for \(\forall _{\alpha } \in {\varGamma }\); then a is of type \(\beta \) for \(\exists _{\beta }\in \bigtriangleup \). If every token \( a \in A\) satisfies \(\langle {\varGamma },\bigtriangleup \rangle \); then \(\langle \) \({\varGamma }\),\(\bigtriangleup \) \(\rangle \) is called a “constraint” supported by \( A\), and denoted as \({\varGamma }\mathrm {\vdash }_A \bigtriangleup \).

A local logic: \(\mathcal {L} = A\),\(\mathrm {\vdash }_{\mathcal {L}}\), \(\mathrm {N}_\mathcal {L}\) consists of a classification A, a set \(\mathrm {\vdash }_{\mathcal {L}}\) of sequents of \( {A}\) called the constraints of \(\mathcal {L}\), and a set \(\mathrm {N}_\mathcal {L}\subseteq tok(A)\) of tokens called the normal tokens of \(\mathcal {L}\), which satisfy all the constraints of \(\mathcal {L}\).

\(\mathcal {L}\) is sound if \(\mathrm {N}_\mathcal {L}=tok(A)\). \(\mathcal {L}\) is complete if \(\mathrm {\vdash }_{\mathcal {L}}\) includes all the constraints supported by \( {A}\). Given a classification \( {A}\), a sound and complete local logic, called Log(A), is generated from \( {A}\).

An infomorphism is important relationship between two classifications and provides a way of moving information back and forth between them. Infomorphism \(\langle \mathrm { f } ^\vee , \mathrm { f }^\wedge \rangle \) is a pair of functions, in which \(\mathrm { f } ^\vee \) is a function from the types of one of these classifications to the types of the other, and \(\mathrm { f }^\wedge \) is a function from the tokens of one of these classifications to the tokens of the other. Given two classifications \( {A}\) and \( {B}\), an infomorphism from \( {A}\) to \( {B}\) written as \(A \rightleftharpoons B\) satisfies the following Fundamental Property of Infomorphisms:

$$\begin{aligned} \mathrm { f } ^\vee (b)\; \mathrm {\models }_A \;\; \mathrm {iff} \;\; b \mathrm {\models }_B \; \mathrm { f }^\wedge (\alpha ) \end{aligned}$$
(1)

for each token \(b\in ~tok(B)\) and each type \(\alpha \in typ(A)\). Relationship between two classifications, \( {A}\) and \( {B}\) is shown in Fig. 1.

Fig. 1.
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Infomorphism of between classification \( {A}\) and classification \( {B}\)

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An information channel: \(C = \mathrm { f } _i : \mathrm {A} _i\rightleftharpoons C\), where \( i \in I\), is an index family of infomorphisms with a common codomain C called the “core the channel.” \( {I}\) is an index set.

2.3 Chu Spaces

Category theory (CT) has been provided a unified language for managing conceptual complexity in mathematics and computer science. Barr used Channel Theory as models of linear logic. Subsequently, Pratts has applied Chu space on variety of mathematical objects [13].

A Chu space \( {A}\) over a set \( {S}\) is a triple (A, r, X), consists of a set A of tokens, a set of X of types, and a function \(r : A \times X \rightarrow S \) gives a binary relation, and S is a set consists 0 and 1. In the Chu space context, tokens are usually called points, while types are called states. It becomes possible to represent a variety of structured objects.

Let \(\mathbf{A}=(A,r,X)\) and \(\mathbf{B}=(B, s, Y)\) be two Chu spaces. A Chu transform from \(\mathbf{A}\) to \(\mathbf{B}\) is a pair (f, g) consisting of functions \(f:\mathbf{{A \rightarrow B}}\) and \(g: \mathbf{{Y\rightarrow X}}\) such that \(s(f(a),y)=r(a,g(y))\) for all a in \(\mathbf A\) and y in \(\mathbf{Y}\). It can be seen that the notion of classification is corresponding with the notion of Chu space and also an infomorphism is a kind of Chu transform.

3 A Model of Collaboration Mechanism

In order to clarify the way to produce an effective collaboration, the mathematical model of collaboration mechanism is proposed based on Channel Theory. The outline of proposed model is shown in Fig. 2. Assume a situation in which two members who have different academic backgrounds work jointly in a group. Each solid circle indicates a set of knowledge held by a member. Due to the different disciplines, each knowledge is different with another. That is why the two circles do not overlap entirely with each other. Due to synergetic effects in the collaboration, team performance cannot be calculated as a simple union of the abilities of each member \((A \cup B)\). Possible knowledge domain of the team can be indicated as grey area \((R-(A \cup B)\). This situation can be represented by using classification of Channel Theory as shown below the circles in the Fig. 2. Here, we can deduce the knowledge, which can be obtained from synergetic effects by using infomorphism. By using this scheme, we can evaluate an effect of a collaboration by representing what new knowledge can be gotten from the collaboration.

Fig. 2.
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A representation model of collaboration mechanism

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3.1 Example of Collaboration Model in Design Workshop

To verify the ability of the proposed model, an example of collaboration in a design workshop is used for a case study. The collaboration between engineering course students and entertainment media course students is discussed. Knowledge of engineering course students are performed as classification of engineering knowledge. Meanwhile, classification of entertainment media knowledge shows knowledge of entertainment media course students.

Classification of Engineering Knowledge (A). A classification of engineering knowledge can be described as a classification as following:

Each token stands for an information technology such as AR (Augmented Reality) code, GPS (Global Positioning System) and Voice commands respectively. Each type stands for a property of the technology. Each type stands for a type of communication. Type of “Information pull” means information is provided when the user requested it. Type of “Interaction” means information is provided through interaction with the system. Type of “Information push” means the system give the user a notification when information is available. For instance, voice commands is suitable for interactive information pull system. The classification can be represented as a Chu map shown in Fig. 3 (A).

Fig. 3.
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A model of collaboration between engineering course student and entertainment media course student

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Classification of Entertainment Media Knowledge (B). Requirements of users are indicated as entertainment media knowledge. The classification of entertainment media knowledge is described as a classification as following:

Here, each token stands for status of a user. Each type stands for requirement of the users. For example, older declines visualization and they need simplicity design. The classification can be represented as a Chu map shown in Fig. 3 (B).

Infomorphisms from a to B (I). An infomorphism from A to B is derived as shown in Fig. 3 (I). Eventually, the collaboration between them can be represented as matrices shown in Fig. 3 by using the proposed method. The model consists of three classifications, i.e. A, B, and I. Each line in the matrix (I) means a combination between a token in classification A and a token in classification B. For example, “AR code” in (A) is combined with Older in (B) because the first line of (I) has the same element of the first line in (B). While, each column in the matrix (I) means a combination between a type in classification A and a type in classification B. For example, Gameness in (B) is combined with “Interaction” in (A) because the middle column of (I) and (A) have the same element. In this case, infomorphisms are established from “classification of engineering knowledge” to “classification of entertainment media knowledge.” Two infomorphisms have been deduced as shown in Fig. 4.

Fig. 4.
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Infomorphisms from engineering knowledge to entertainment media knowledge

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Each situation explains new knowledge, which engineering course students and entertainment media course students can obtain in the collaboration process. It implies that there are two situations could occur when engineering student and entertainment media student collaborate in a workshop. First infomorphism shows that

This infomorphism shows us that AR code is corresponding to elder, GPS is corresponding to adult and Voice commands technology is corresponding to child. These results provide us a new knowledge for selecting proper information technology device in accordance with the user’s generation, i.e. we should provide AR code technology for elder, GPS for adult and Voice commands for child. Meanwhile, second infomorphism shows that

From this infomorphism, AR code technology can be implied as same as infomorphism 1. But GPS is corresponding to child and Voice commands is corresponding to adult.

From the above discussion, we can say that the collaboration is effective because it can provide new knowledge to the members.

4 Case Study: International Design Workshop

To verify the discussion in the previous section, an international design workshop was held in Hokkaido, Japan from 25–28, October 2014. The participants were students from engineering courses of universities in Japan and students from an entertainment media course of an university in Thailand. The theme of this workshop is “Enhance tourism.” The schedule of the workshop is described as follows:

Collecting Before Data. Before start collaboration process, a questionnaire was conducted in order to observe current knowledge of each member. Participants were asked to answer the questionnaire within 30 minuets. The questionnaire was open questions in which participants are asked to describe their ideas about how to enhance tourism in Hokkaido by describing as keywords, methods and expected results.

Survey. During the workshop, members were instructed to observe many famous sightseeing places in Hokkaido, Japan. The instructor assigned participants to take photographs of interesting things that they found by using Category cam application. Category cam application is an application which helps users to take photographs together with a short note. After a user took a photograph, he/she can select labels from Kansei (Emotion), Cultural, Physical and Other, that explain the photograph accurately. Then, the user can give a short note about the photograph. By using this application, a user can record their impression easily. Snapshots of screen of Category cam application are shown in Fig. 5.

Fig. 5.
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Snapshots of screen of category cam application

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Group Work and Presentation. After the survey step, members were divided into four groups. Each group was comprised students of engineering course and students of entertainment media course. Members made group discussions with own team members to propose a project to enhance tourism in Hokkaido. During the group discussions, KJ method [14] was conducted. KJ method is a fundamental tool to organize ideas and data through a brainstorming. During the process of KJ method, all members were sharing their observed information by using the photographs and short note, which were taken in the survey stage. Then, they worked together to create a project to develop tourism in Hokkaido, Japan. In the last day of the workshop, each group made a presentation about their proposal. Scenes of the workshop are shown in Fig. 6.

Fig. 6.
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Scenes of the international design workshop

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Collecting After Data. After the presentation was finished, the participants were asked to answer the questionnaire again to investigate how the knowledge of each member changed. The questions in the second questionnaire was same as the first one. Examples of completed questionnaire sheets are shown in Fig. 7. Important keywords were picked up from the questionnaires. The number of the keywords of the first questionnaire and the second questionnaire were compared. The number of keywords of entertainment media course grow 95 % on average. Meanwhile, the number of keywords of engineering course grow 150 % on average. The results are shown in Fig. 8 as bar graphs. From the results, we can see that participants can enlarge their knowledge from the collaboration. Consequently, we can say that this workshop could make effective collaboration.

Fig. 7.
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An example of a set of questionnaire of a member

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Fig. 8.
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The number of keywords which have gotten from participants by comparing between the first and second questionnaires

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5 Conclusion

In this paper, the authors have represented a model of collaboration mechanism based on Channel Theory. Collaboration between engineering course students and entertainment media course students was represented by using the proposed model. In the example, classification \( A\) stands for knowledge of engineering student and classification \( B\) stands for knowledge of entertainment media student. As the results, there are two infomorphisms were deduced from the classifications. It means the students could gain new knowledge from the collaboration process and we can say that the collaboration is effective.

To verify the effect of the collaboration, a workshop was conducted. The participants were student from engineering courses of university in Japan and students from an entertainment media course of university in Thailand. During the workshop, the participants were asked to answer the questionnaires to investigate how the knowledge of each member changed. The results show that the participants can enlarge their knowledge from the collaboration.

The proposed model can represent new knowledge which members can obtain from the collaboration. This new knowledge could lead the team to novel solutions. The team performance can be estimated by analyzing the model of the team. Moreover, it is expected that the proposed model can be employed when a new team is organized. The team manager can use the model as a decision supporting tool for organizing a team.