Measuring open innovation under uncertainty: A fuzzy logic approach

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Introduction
Since the first work by Chesbrough (2003), the Open Innovation (OI) paradigm has been the focus of much attention.OI is growing greatly in importance both in academia and industry, as well as in the public policy domain (Bogers et al., 2018).It involves firms increasingly relying on external sources of innovation.Hence, the target of OI is both the integration of external sources of innovation in the company (inbound) and the identification of external paths for commercialising internally sourced innovations (outbound) (Cheng et al., 2016).Moreover, it is expected that OI will play a crucial role for businesses to recover faster from Covid-19 disruption (Chesbrough, 2020).Several recent studies have been conducted to analyse the current OI state of the art and to set up future research agendas (Bogers et al., 2017;Chesbrough, 2017;Hossain and Kauranen, 2016;Randhawa et al., 2016), as well as to investigate OI adoption among companies (Radziwon and Bogers, 2019) and to develop systems for effective OI management (Lamberti et al., 2017;Santoro et al., 2018).The increasing amount of knowledge and information flows today requires the development and implementation of digital technologies to manage open innovation processes (Urbinati et al., 2020).
OI is not a clear-cut concept, since it comes in many forms and can be implemented in different ways (Huizingh, 2011).Thus, in order to effectively manage OI, it needs to be properly measured.Nonetheless, its assessment has proved to be highly challenging and a wide variety of measures and approaches have been proposed in the literature.Although a holistic innovation measurement system and integrated framework have been proposed (Dewangan and Godse, 2014), a conceptual framework which proxies the adoption of the OI paradigm by companies under a holistic perspective is lacking (Lamberti et al., 2017).Furthermore, the approaches available in the literature do not take into account the uncertainty of human judgment and the vagueness related to the paradigm.In fact, the measures suggested by scholars are often operationalised in a crisp way, e.g. through dummy variables which are variables assuming only the values of 0 or 1 (see, e.g., Laursen and Salter (2006)).
Fuzzy logic has proven to be one of the most widely used and efficient methods to treat decision making problems affected by uncertainty (Mardani et al., 2015;Mondragon et al., 2019;Petrovic et al., 2006).Fuzzy logic has been used in numerous applications, including, a fuzzy inference system for evaluating green supply chain management performance (Pourjavad and Shahin, 2018), a real option theoretic fuzzy evaluation model for enterprise resource planning investment (You et al., 2012), a fuzzy logic based model for risk assessment in additive manufacturing R&D projects (Cabezali and Santos, 2020), and a fuzzy based system for risk analysis and evaluation within enterprise collaborations (Wulan and Petrovic, 2012).In particular, it can be usefully applied in situations when (1) data values and relations between them are uncertain and imprecise and their estimation is based on the subjective beliefs of individuals; (2) it is difficult to measure data, either because there is no unit of measurement or there is no quantitative criterion for representing their values; (3) parameters that characterise an OI problem are vaguely and unclearly defined; (4) the knowledge available about OI is complex, limited or incomplete; and (5) human reasoning, perception or decision making are inextricably involved in problem solving.
Therefore, based on the considerations above, two research questions were formulated: .
1. How can we build a holistic framework to assess the degree of openness of a company's innovation taking into account the vagueness and uncertainty of the judgment?2. How can we combine the many different qualitative and quantitative factors with different units of measurements affecting the degree of OI?
This work addresses such gaps in three ways: Firstly, on the basis of a comprehensive review of the OI literature, six categories of key aspects to be considered in OI assessment were grouped into three building blocks: enablers, activities and outputs and the corresponding metrics were identified.Secondly, given that OI assessment is a multi-dimensional problem involving several metrics with different units of measurement and that both the data available and the decision maker's reasoning may be affected by vagueness and uncertainty, a novel modular fuzzy rule based system was developed in order to help managers evaluate their company's openness to innovations.Lastly, the practical validity of the fuzzy rule based system devised was tested through three numerical examples and case studies which included the assessment of the OI level of two companies.
Therefore the ultimate research objective of this work was: .
• To develop a fuzzy logic based approach for decision makers to assess the degree of OI in firms.
The remainder of this paper is organised as follows.Section 2 contains the OI literature review and defines the conceptual framework, whilst Section 3 presents the fuzzy logic based OI assessment system.The validity of the system is reported and discussed in Section 3.3 using three numerical examples and two real-world case studies.Finally, conclusions and future research directions are summarised in Section 4.

Literature review
This section aims to provide an overview on the OI assessment problem and describe the gap that this work intends to fulfil, as well as to identify the categories and metrics selected for the proposed framework.

OI assessment problem
Several studies have been conducted with the aim of assessing OI and understanding the factors that have an impact on it.Some works have focused on developing frameworks to assess innovation excellence, such as the value-generating capability of innovation management and the identification of ways to measure and improve it (Dervitsiotis, 2010).Moreover, the assessment of firms' OI readiness and maturity has been a subject of considerable interest.For example, Remneland-Wikhamn and Wikhamn (2011) proposed an Open Innovation Climate Measure (OICM) tool for assessing the organizational climate for successful implementation of open innovation, Enkel et al. (2011) developed an open innovation maturity framework to measure and benchmark excellence in open innovation, Waiyawuththanapoom et al. (2013) proposed a conceptual framework called Open Innovation Readiness Assessment Model (OIRAM) that provides a set of guidelines on how companies can both assess, as well as improve their ability to implement open innovation initiatives successfully.Rangus et al. (2016) devised a firm-level measure of proclivity for open innovation, which relates to the firm's willingness to perform inbound and outbound open innovation activities.Cheng et al. (2018) developed an evaluation tool to include the mindset and behavior intersect in OI activities at organizational level to examine organizations' readiness for OI adoption.de Oliveira et al. ( 2019) assessed the critical success factors to implement OI in small or medium-sized enterprise (SMEs) through a set of indicators.Silva et al. (2019) used cognitive mapping and the Choquet integral to identify and prioritise relevant criteria for evaluating SMEs' propensity for OI.Nestle et al. (2019) measured open innovation culture in terms of not-invented-and not-sold-here syndromes, investigating the role of trust and information asymmetry.
Other authors have focused their attention on assessing OI performance, such as Tsai and Liao (2014) who devised a framework for open innovation assessment based on dimensions extracted from the literature.Rogo et al. (2014) proposed a methodology for assessing the performance of OI practices, with a focus on Intellectual Capital.Al-Belushi et al. (2018) developed a quantitative measure for open innovation to determine the performance of a firm within the marine biotechnology sector, with a focus on the two OI dimensions of breadth and depth.
Regarding the assessment of the OI degree of firms, Michelino et al. (2014) proposed a methodology for measuring companies' openness through accounting data and providing a relation between open innovation activities and intellectual capital components.Michelino et al. (2015) devised a framework for measuring the pecuniary dimension of OI in both inbound and outbound processes, through the analysis of annual reports of bio-pharmaceutical companies.The paper by Carroll et al. (2017) proposed a practical dashboard for assessing OI approaches in the context of early drug discovery.Lamberti et al. (2017) proposed a scorecard with a suite of indicators that provided innovation managers with a measure of OI adoption in their organisations.Furthermore, Rosa et al. (2020) focused on measuring OI practices in small Brazilian companies by means of a set of indicators.Recently, Marullo et al. (2021) developed a framework based on Item Response Theory to understand the level of firms' openness, conducting a large survey of technology-based SMEs.
This analysis of the literature showed that the majority of the studies were focused only on investigating certain dimensions affecting the degree of OI.Although other works proposed a more comprehensive assessment framework, including multiple factors, these approaches were based on measures often operationalised as crisp variables and did not consider the uncertainty and the vagueness of human judgment.Therefore, little attention has been given to developing a holistic system including all the main factors affecting the degree of OI, able to handle variables with different units of measurements and reflecting the uncertainty related to human judgement.This work attempts to fill this gap by devising a model based on fuzzy logic for measuring the degree of a firm's OI.

OI criteria and metrics selection
The bibliometric search was intended to cover studies on OI metrics.We employed a two-stage approach to select a comprehensive set of scholarly publications that form the basis of our review.Firstly, a database search using the key term "open innovation" was conducted.The EBSCO database was employed in order to collect the main body of literature for the analysis and the search was restricted only to journal articles including "open innovation" in the title, keywords or abstract.Drawing on this sample of studies, our second step consisted of reading the articles and following footnotes and references to other articles and books.Such an approach led to the inclusion of other contributions.
Based on the aim of this review, three criteria were adopted to define a set of comparable studies: 1. "Open innovation" was searched for, so as to only include papers referring directly to the paradigm.2. Studies which do not explicitly make reference to OI were omitted.Thus the timeframe was restricted to articles published from 2003 onwards.
Fig. 1.The conceptual framework for OI assessment.
3. The included contributions comprised the analysis of OI adoption, since the primary focus of this review was to better understand which indicators were used to assess a firm's degree of openness.
On the basis of the review, a conceptual framework was built which interprets the key aspects scholars took into account for OI assessment.In particular, six categories of metrics grouped into three building blocks -enablers, activities and outputs-were identified, as shown in Fig. 1.
The enablers of OI include two categories: outer environment and internal climate.
The former refers to the fact that external environment plays an important role, since OI breaks down company boundaries and involves active interaction with external entities.As such, the indicators belonging to this category mainly aim to: .
The latter regards the internal climate of a company, reflecting the fact that moving from a closed to an open way of innovating is not easy and often involves significant cultural and organisational change.For instance, the "not invented here" syndrome is one of the greatest obstacles to the implementation of OI and occurs when members of a firm are resistant to ideas, technologies and knowledge coming from outside (de Araujo Burcharth et al., 2014).Therefore, creating an appropriate culture (Nestle et al., 2019), developing specific capacities (Ahn et al., 2013), using internal processes, structures, systems and tools (Zhang et al., 2018), predisposing supportive organisational units (Lazzarotti and Manzini, 2009) and accessing the right skills embodied in new personnel (Santamaria et al., 2010) have deserved particular attention in metrics falling into this category.Moreover, at this level, Remneland-Wikhamn and Wikhamn (2011) built a three-dimensional assessment tool to evaluate OI climate.
The activities performed for OI comprise three categories, the first two concerning the authorship of innovation (collaboration and importing mechanisms) and the third pertaining to appropriability (protection mechanisms).As to authorship, a distinction is made between collaboration (joint authorship) and importing mechanisms (external authorship), both reflecting strategies engendered by the OI model and contrasting with internal authorship, i.e. the closed model of innovation.As a matter of fact, innovations can be developed mainly by: 1) the firm together with other enterprises or institutions, through collaboration; 2) other enterprises or institutions, through acquisition; or 3) the enterprise or enterprise group, through in-house research departments (Acha, 2008).
Hence, the indicators belonging to the collaboration category primarily aim to: .
As to importing mechanisms, the research focus has been principally based on the acquisition of external R&D, knowledge and innovation by means of outsourcing (Brunswicker and Vanhaverbeke, 2015), technology purchasing (Lee et al., 2010) or in-licensing (Henttonen and Lehtimaki, 2017), also detecting the costs and the investments related to such transactions (Michelino et al., 2014).
Regarding protection mechanisms, it should be noted that although a tight appropriability regime can be perceived as a closed innovation strategy, the registration of intellectual property may also be used as a tool to commodity proprietary knowledge, potentially facilitating greater interaction.In this sense, the metrics falling into the protection mechanisms category are conceived on the assumption that appropriability is a crucial practice in securing positive economic returns from the outbound process (Manzini and Lazzarotti, 2016).The actions that a firm may take to regulate knowledge outflow (Greco et al., 2015) also belong to this category.
Lastly, the outputs denote the results achieved through OI.Indeed, opening firms' boundaries to let innovations flow outside may generate both operational and financial benefits (Michelino et al., 2014).As such, the exporting mechanisms refer to activities where internal know-how, products or innovations are monetised on an external market.Thus, such indicators mainly estimate: .
• the new revenue opportunities derived from licences, spin-off and sales divestiture (Chesbrough, 2006); • the models companies apply when trading in technology (Podmetina et al., 2011); • the contribution to the general public when revealing knowledge for free (Carlsson et al., 2011).
Table 1 shows the six categories (henceforth, dimensions) and the corresponding indicators (henceforth, sub-dimensions) identified above, together with a set of metrics for their measurement.
However, the operationalisation of the OI metrics available in the literature made use of different approaches based on dummy, percentage and Likert scale variables, which might not be suitable for modelling real situations affected by uncertainty and vagueness.Therefore, the aim of this work is to make use of fuzzy logic in order to take uncertainty and human reasoning into account when assessing a firm's degree of openness.

Fuzzy OI assessment system
The OI assessment proposed is structured in 3 blocks: Enablers, Activities and Outputs which include 6 dimensions, 13 subdimensions and 19 OI metrics, as presented in Table 1 and the previous section.Each of the defined blocks, dimensions, subdimensions and OI metrics, are represented as linguistic variables and modelled using fuzzy sets.The relationships between them are modelled using fuzzy If-Then rules and the assessment carried out is based on a fuzzy inference mechanism.each of them within the interval of 0-1, where 0 represents a complete non-membership and 1 complete membership to the fuzzy set.Fuzzy logic is based on the concept of linguistic variable and membership function.A variable such as OI may assume certain linguistic values (for example, Very Low, Low, Medium, High, Very High).A membership function for each linguistic value is defined in order to express the degree of membership of each value in interval [0,1] to the linguistic variable, as shown in Fig. 2. For instance, within a minimum value of 0 and a maximum of 1, an OI index of 0.4 might be considered to belong to the fuzzy sets corresponding to the linguistic variables Low and Medium, with different degrees of membership, namely 0.4 and 0.6.Membership functions can assume different shapes depending on the particular problem and variable, but the most commonly used one is triangular.It is represented by a triplet of numbers corresponding to the x-axis coordinates of the triangle's vertexes, where the lowest, the middle and the highest x-axis coordinates have membership degrees of 0, 1 and 0, respectively.

Notes on fuzzy sets and fuzzy inference systems
Fuzzy logic finds its natural implementation in fuzzy inference systems (FISs) (Mamdani and Assilian, 1993;Takagi and Sugeno, 1985).An FIS is a system which maps certain input linguistic variables to a certain output using If-Then rules.As input, it takes crisp data, which are then fuzzified through a fuzzification process consisting in the computation of the degree of membership of the input crisp data with respect to the corresponding linguistic variables in the If part of the rules.Two different fuzzy inferences were defined: Mamdani (Mamdani and Assilian, 1993) and Sugeno (Takagi and Sugeno, 1985).These methods differ in the way they calculate the output: the former defines a fuzzy set, while the latter uses a single scalar as a membership function of the rule consequent.Mamdani fuzzy inference has proven to be a valid approach for capturing expert knowledge and is selected for the proposed OI assessment system where the knowledge represented in the Then part of a rule is also uncertain.Fig. 3 exhibits the proposed FIS for OI assessment receiving as input variables relevant to the three building blocks -enablers, activities and outputs-.
Once input data have been fuzzified, a fuzzy reasoning process takes place by firing rules defined in order to relate the fuzzy input to the fuzzy output.Rules emulate human reasoning and the decision-making process of experts.In the case of two input variables, one output variable and three linguistic terms (Low, Medium and High) for all the variables, rules may be formulated, for example, as follows:

If External R&D is Low and Inbound acquisition is Low Then Importing mechanisms is Low If External R&D is Low and Inbound acquisition is Medium Then Importing mechanisms is Low If External R&D is Low and Inbound acquisition is High Then Importing mechanisms is Medium If External R&D is Medium and Inbound acquisition is Low Then Importing mechanisms is Low If External R&D is Medium and Inbound acquisition is Medium Then Importing mechanisms is Medium If External R&D is Medium and Inbound acquisition is High Then Importing mechanisms is High If External R&D is High and Inbound acquisition is Low Then Importing mechanisms is Medium If External R&D is High and Inbound acquisition is Medium Then Importing mechanisms is High If External R&D is High and Inbound acquisition is High Then Importing mechanisms is High
Fig. 4 shows an example from the system built in Matlab of a two input Mamdani fuzzy inference system with 9 rules.External R&D and Inbound acquisition are fuzzified by finding the intersection of the crisp input values with the corresponding input membership function.Then, the minimum of the two input membership values is used to obtain a rule strength which is used to cut the Importing mechanisms output membership function at the rule strength.Finally, the outputs of the fuzzy rules are combined into one output fuzzy membership function by taking the maximum of the 9 output membership values.
Once an output membership function has been calculated, a crisp value which best represents the corresponding fuzzy set needs to be determined through a defuzzification process.Several defuzzification methods are suggested in the literature.However, none of the Fig. 3. FIS for OI in function of the three building blocks.

E. Mastrocinque et al.
defuzzification methods is the best and the selection depends on a particular application.The method we used in this study was the Mean of Maximum, also used in many other systems (Carrera and Mayorga, 2008).The Mean of Maximum calculates the mean of all the values which reach the maximum membership value in the corresponding fuzzy set, as shown in Fig. 4.

Structure of the proposed fuzzy rule based system for OI assessment
All FISs defined for the OI assessment were connected together in a modular approach where the output generated by each FIS, used to assess the OI at one level, was the input to another FIS which assessed the OI at the subsequent level, as shown in Fig. 5.The proposed modular fuzzy system was composed of 4 levels, 13 sub-systems and 19 input variables.The first level corresponded to the OI computation based on the assessment of the 3 building blocks, the second one to the assessment of the 3 building blocks based on the 6 dimensions, the third represented the assessment of the 6 dimensions based on the 13 sub-dimensions, and, finally, the fourth corresponded to the assessment of the 13 sub-dimensions based on the input OI metrics.Importing mechanisms and Exporting mechanisms were the only dimensions with one sub-dimension each and consequently they received their respective variables directly as input.Some sub-dimensions, such as Human resources, Partners, Acquisition of external knowledge and Protection methods, were computed using more than one variable and for this reason four sub-systems were added to the model in the fourth level.Moreover, the remaining sub-dimensions were estimated by a single variable, and for this reason were given directly as an input for the sub-systems in the third level.
Each sub-system is tasked with calculating a single output such as a sub-dimension, a dimension, a building block and the overall degree of OI.This modular approach enables the decision maker to keep all the factors responsible for the final OI assessment under control, and, as a consequence, to understand where it is possible to act in order to improve the firm's degree of openness.
Subsequently, rules were defined in order to link the input variables to the output ones for each fuzzy sub-system.The same weight of 1 was assigned to each rule.The sub-systems had a different number of rules depending on the number of input variables.Three rules were defined when there was only one input variable.For a two-input sub-system 9 = 3 2 rules were defined, as shown in the previous sub-paragraph example.A maximum number of rules 27 = 3 3 was defined for a three-input sub-system; for example,  The advantage of using a modular approach is the reduction of the number of rules.Interestingly, if a single fuzzy system with 19 input variables with 3 linguistic values each was to be defined, it would lead to the definition of 3 19 = 1, 162, 261, 467 rules.

If Partners is Low and Forms is Low and Innovation funnel is Low Then Collaboration is Low If Partners is Low and Forms is Low and Innovation funnel is Medium Then Collaboration is Low If Partners is Low and Forms is Low and Innovation funnel is High Then Collaboration is Medium If Partners is Low and Forms is Medium and Innovation funnel is Low Then Collaboration is Low If Partners is Low and Forms is Medium and Innovation funnel is Medium Then Collaboration is Medium If Partners is Low and Forms is Medium and Innovation funnel is High Then Collaboration is Medium If Partners is Low and Forms is High and Innovation funnel is Low Then Collaboration is Medium
Finally, the Mean of Maximum defuzzification method was used in all the sub-systems.This method prevented the values of the variables being attenuated while they were reaching the final level of the system (Fig. 5).

Linguistic variables in the proposed system
Crisp variables used in various deterministic OI assessments employ different approaches such as dummy, percentage and Likert scale variables.In the proposed fuzzy system, each metric was defined as a linguistic variable and fuzzified based on the operationalisation available presented in the literature.Furthermore, a questionnaire was designed to collect the values of the input variables to validate the proposed system and to assess the firms' degree of openness.
For instance, Governmental support was converted into a continuous metric from 0 to 1 representing the ratio between government grants and R&D expenditure sustained by the firm.Other variables, such as Not shared here, in accordance with the literature, were assessed using an index from 1 to 7 corresponding to the average rate given to the appropriate items of the questionnaire.Moreover, indicators such as Proportion of science and engineering graduates in the workforce, having been proxied as a percentage in the  literature, were still measured as a percentage from 0 to 1. Finally, to assess the sub-dimension "Forms", a variable Alliances was defined and fuzzified as a number from 0 to 5 indicating how many of the listed kinds of collaboration, the company had stipulated.

Implementation
The model was implemented in MATLAB/Simulink environment.Each sub-system was designed using the Fuzzy Logic Toolbox of MATLAB, widely used in the literature (MahmoumGonbadi et al., 2019).Linguistic variables, linguistic terms, membership functions and rules of each sub-system were defined subsequently, and linked following the proposed framework using Simulink, shown in Fig. 5.

System validation
In order to validate the developed fuzzy system, three numerical examples and two real-world case studies were investigated.

Numerical examples
We randomly generated the values of the input variables and defined three numerical scenarios corresponding to different levels of OI adoption by a firm (Tables 3 and 4).
In the first example, although the company performed both collaboration and protection activities to a medium extent, the system calculated a low degree of openness.This is mainly due to not exploiting the innovation opportunities offered by the external environment, the weak embeddedness of OI throughout the organisation and the absence of externalisation strategies for internal innovations.
In the second example, the high score of enablers -near the threshold -indicated that the firm actively seized the opportunities arising in the outer environment for its innovation processes and had also created conditions that were conducive to OI.Moreover, it acquired external knowledge and commercialised internal know-how, products and innovations, even if only to a medium extent.In this instance, an average degree of openness was obtained.
The significant level of OI adoption in the third example derived from the combination of high scores for activities and outputsboth near the threshold -with a medium score for enabling drivers.The company performed all the open activities to a high extent and successfully transferred the outcomes of such activities to the market.In doing so, it drew on external knowledge sources to fuel its innovation activities and relied on a supportive organisational climate.

Case studies
Two different companies were used to test the developed fuzzy inference system for assessing their degree of OI.Both companies were based in Spain.The main business of the first company (Company A) is the manufacture of machinery for the food industry, with 185 employees and an approximate turnover of 32 million Euros.The second company (Company B) deals with recycling of waste and currently has 30 employees and a turnover of about 4.6 million Euros.
A questionnaire was emailed to the managing directors of the companies in order to collect the values for the OI metrics to feed into the fuzzy system.The values received are reported in the Table 5, while the outputs of the system are shown in Table 6.
Both companies exhibit an average level of Enablers due to a fair level of Outer environment and External Climate as well as a low level of Outputs due to the absence of externalisation strategies for internal innovations.However, while Company A performs both collaboration and protection activities to a medium extent, as shown by an average level of Activities and, consequently, of the degree of OI, Company B exhibits a low level of Activities mainly due to a very low use of Protection mechanisms resulting in an overall low degree of OI.Moreover, both companies present a low level of Importing and Exporting mechanisms, showing a closed attitude towards acquiring and selling know-how and innovation.
Overall, the results obtained by using the fuzzy system proposed are unsurprising due to the different nature of the businesses, as well as the size and turnover of the companies.In fact, larger companies are expected to have a higher degree of OI than smaller ones.
However, both companies show either low or medium values for the dimensions, building blocks and degree of OI, while no high values were found.This is because the majority of companies are still yet to adopt strategies fostering a high level of OI showing a disparity between theory and actual practice.
Such findings clearly lead to the practical implications of the developed system.It can support managers in assessing their company's overall openness in terms of external and internal environmental factors, development, protection and externalisation strategies.They can use their subjective knowledge to identify the areas of improvement and to make informed decisions regarding adoption of strategies to increase their degree of OI.
However, the use of fuzzy logic is not without limitations.In fact, the accuracy of the results are subjected to the definitions of the variables' intervals, membership functions and rules.Therefore the proposed model, although including all the main factors contributing to OI, represents a baseline regarding the adoption of fuzzy logic for assessing OI.

Managerial implication
The proposed framework and fuzzy inference system have significant managerial implications.Firstly, the main dimensions, subdimensions and metrics to measure a company's degree of openness are provided to decision makers.Moreover, fuzzy logic has been adopted to include the uncertainty and vagueness of the decision maker's judgement.Secondly, the system has been conceived and designed to provide an easy, fast tool for solving such a complex assessment problem.The modular FIS will allow the decision maker to assess the different areas contributing to the overall degree of openness of the company, and to identify both strong and weak areas.Finally, the framework will allow decision makers to explore different routes to increase the overall OI level.

Concluding remarks
This paper sought to develop a fuzzy logic based system for OI degree assessment.Through a comprehensive literature review, we developed a six-factor framework comprising meaningful categories of OI metrics qualified in terms of three building blocks -enablers, activities and outputs.Fuzzy sets theory was then used as a tool to build an OI assessment system, allowing managers to estimate the openness of their firms from a holistic perspective.The practical validity of the system was tested through the assessment of three numerical examples and two real-world companies, showing different OI levels.
The findings of this work provide an answer to the research questions formulated in Section 1, building an inclusive framework for OI assessment under vagueness and uncertainty.It does so by combining several different qualitative and quantitative factors with different units of measurements, using fuzzy logic and fuzzy inference systems theory.Moreover, the real-world case studies used to test the applicability of the system show that both companies could do more in terms of Importing and Exporting mechanisms and, overall, being more open to innovation.The case study of the two companies is by no means exhaustive regarding the degree of OI in industry, but they might be considered as representative of the actual level of OI strategy adoption in many organisations.
This study offers managers both general principles in the development of OI metrics as well as a sample of specific indicators which they can use to comprehensively track the innovation openness of their companies.Moreover, it identifies a set of areas to be measured in order to gain an insight into an organisation's holistic ability to manage OI.Once the degree of openness is assessed, the potential emerges to explore the relationship between the level of OI adoption by firms and their innovative and financial performances.The conceived OI framework and the developed modular fuzzy inference system represent a substantial contribution to the OI literature and help feel the gaps regarding the problem of OI assessment while also proposing a novel and interesting application of fuzzy logic.
Future research directions are threefold.Firstly, a survey comprising a large pool of companies will help confirm and extend the validity of the proposed OI assessment framework.Secondly, the proposed fuzzy inference system can be implemented in companies' information systems as well as a cloud-based tool in order to help managers in the assessment, monitoring and improvement of their firms' OI level.Finally, a fuzzy logic based system for measuring OI performance may be devised.

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.Mastrocinque et al.
Mastrocinque et al.If Partners is Medium and Forms is High and Innovation funnel is Low Then Collaboration is Medium If Partners is Medium and Forms is High and Innovation funnel is Medium Then Collaboration is Medium If Partners is Medium and Forms is High and Innovation funnel is High Then Collaboration is High If Partners is High and Forms is Low and Innovation funnel is Low Then Collaboration is Medium If Partners is High and Forms is Low and Innovation funnel is Medium Then Collaboration is Medium If Partners is High and Forms is Low and Innovation funnel is High Then Collaboration is Medium If Partners is High and Forms is Medium and Innovation funnel is Low Then Collaboration is Medium If Partners is High and Forms is Medium and Innovation funnel is Medium Then Collaboration is Medium If Partners is High and Forms is Medium and Innovation funnel is High Then Collaboration is High If Partners is High and Forms is High and Innovation funnel is Low Then Collaboration is Medium If Partners is High and Forms is High and Innovation funnel is Medium Then Collaboration is High If Partners is High and Forms is High and Innovation funnel is High Then Collaboration is High Overall, 231 rules were defined. E.

Table 2
List of input variables and their fuzzy representation.

Table 3
Values of the input variables for the three examples.

Table 4
Results of the modular fuzzy system for the three examples.

Table 5
Values of input variables received from Company A and B.

Table 6
Results of the modular fuzzy system for Company A and B.