**3. Methodology**

In order to fulfill the objective of the research and to prove the model showed in Figure 1, we implement a fuzzy inference evaluation system, thus evaluating the degree of relationship between [11] managemen<sup>t</sup> control indicators and financial and non-financial performance. This is done by generating a fuzzy logic analysis, whose model is coupled with MATLAB's software to take advantage of the Fuzzy Logic Toolbox tool. The application of this was done on Chilean SMEs, and the fuzzy inference system (FIS) is based on [28] and the Mamdani method for fuzzification [13].

#### *3.1. Fuzzy Inference System in Management*

The origins of fuzzy logic stem from [29–31] research, Professor at the University of Carolina (USA), in his article "Fuzzy Sets". Zadeh proposed a mathematical framework for imprecise data, breaking paradigms by shifting from Boolean logic (0–1; white–black, true-false) to fuzzy logic, which implies that the elements belong to a set to a certain degree. As a consequence, a large variety of greys emerged from the traditional black and white [32,33]. In the field of social sciences, fuzzy logic delivers managemen<sup>t</sup> techniques in an environment that has imprecision, uncertainty, incomplete information, conflictive information, truth bias, and possibility bias [34]. When discussing fuzzy logic, it must be understood that the basic underlying concepts are linguistic variables, which are variables whose values are expressed by words and not numbers. In effect, fuzzy logic must mainly be seen as a methodology to calculate words instead of numbers [28].

Fuzzy inference systems (FIS) are methodologies that express knowledge and inaccurate data, which is very representative of human thought. Therefore, this method is best employed to give answers to problems that have latent variables instead of observed variables [35]. It defines a non-linear relationship between one or more input variables and an output variable. This provides a starting point for decision-making to take place [36]. Phases:


#### *3.2. Mamdani-Fuzzy Rule Type-Based Modelling*

There are different fuzzy inference models, and their use depends on the type of problem that needs to be solved. The main difference between models lies in the consequences of the rules and in the aggregation and fuzzification methods [28]. Consequently, the research applies Mamdani's [13] model because the inputs and outputs are linguistic rules. This investigator used [29] proposal as a base regarding fuzzy algorithms for complex systems and decision-making processes [28].

Mamdani's model proposes the IF-THEN rules. This implies a series of rules. Input (regressions) matrix and as an output vector are defined as follows:

$$\begin{aligned} \mathsf{X} &= [\mathsf{x}\_{1\prime} \; \dots \; \mathsf{x}\_{2}]^{\mathrm{T}} \left[ \begin{array}{c} \mathsf{X}\_{11} & \mathsf{X}\_{12} \\ \mathsf{X}\_{21} & \mathsf{X}\_{22} \\ \mathsf{X}\_{n1} & \mathsf{X}\_{n2} \end{array} \right] \\ \mathsf{G} &= [\mathsf{g}1 \; \dots \; ... \; \mathsf{g}n] \end{aligned}$$

Hence, the Mamdani fuzzy model is made of fuzzy propositions in its antecedents and consequents. The general rule is IF-THEN [39]:

Ri: if x is Ai then y is Bi; i = 1; 2; . . . ; K

Ri is the rule number.

Ai and Bi are the fuzzy sets.

x is the antecedent variable representing the input in the fuzzy system.

y is the consequent variable related to the output of the fuzzy system.

The study employs triangular fuzzy membership and trapezoidal fuzzy membership functions. This study elaborates fuzzy inference for each control lever as proposed by [11], also analyzing the distance that exists between their use and performance in SMEs.

Measuring the use of the managemen<sup>t</sup> control tools in SMEs is a difficult and complex task to convert into quantitative values as they are partially composed of qualitative data. This implies the need to measure multiple attributes such as the existence of a mission, vision, values, holding meetings, and definition of organizational structure, among others. Hence, FIS serves as a reliable tool to tackle uncertainty in an environment rife with imperfect information.

The analysis also implies measuring the "performance" variable in a company, taking into account a variety of indicators that are impossible to measure at a unidimensional level.

Mamdani's model is built by considering a series of linguistic proposals and by elaborating different rules measured from observed data. Information is obtained by means of a survey structure that proposes a measuring tool for the levers through fuzzy inference systems.

The software employed in the research is MATLAB–Fuzzy Logic Toolbox. These are grouped in the following levers: beliefs, boundary, interactive, and diagnostics. "Performance" is also measured by considering Financial (liquidity, debt, capability to pay providers, utility) and Non-financial (organizational environment, sense of business control, improved decision making) aspects.

## *3.3. Sample Characteristics*

The target population of this study is composed of Chilean SMEs (medium: 51 to 200 workers; small: 1 to 50 workers) that are part of the 2018 database of the Federation of Chilean Industry SOFOFA (Sociedad de Fomento Fabril) published with 4000 enterprise members. In Chile, SMEs represent 96.8% of the total companies in the country, 220,000 are SMEs, and nearly 680,000 are small businesses.

Final sample size n = 86, with a response rate of 7.1%. The sampling used is not probabilistic, which limits the conclusions of this article.

Table 1 summarizes the descriptive statistics of the surveys. This research employed SPSS to code the survey and to elaborate an exploratory factorial analysis, which is shown in the first part of this investigation.

**Table 1.** Profile of sample companies.


From the previous table, we can identify that the majority of companies that answered the survey are of small size, with a mode of 10 employees, and are mostly from the commercial sector of the Metropolitan Region.

Of the individuals who responded to our survey, the average work experience is 16 years, and 46% have higher education. Most of the surveys were answered by the CEO; hence, it can be foreseen that they have a high degree of knowledge of their companies.

#### *3.4. Construction of the Variable Measurement Survey*

To measure the variables in the study, we developed a structured survey to collect the empirical data. The survey design first required an exhaustive literature review on the Web of Science–Clarivate Analytics database. Following the review of published works, no empirical study was found pertaining to these variables and scales intended to measure the constructs. Hence, a proposed measuring scale for each dimension in the research is presented in Table 2. The validity of the content of the initial survey proposal was evaluated by experts in the field of managemen<sup>t</sup> control and market research, with an additional random sample of 10 SMEs to evaluate content. An exhaustive review was elaborated to achieve a good level of acceptance of the survey in its draft, number of questions, and

design. We received comments that helped us improve the design of the survey, focusing on internal validity and content, giving us an acceptable measuring scale. This allowed us to attain internal coherence for all the dimensions in the study model.

**Table 2.** Measurement Scales.


Source: Prepared by the authors.

The measurement scale for the dimensions is managemen<sup>t</sup> controls, human resources, and organizational performance measured with a Likert scale from "1" (strongly disagree) to "5" (strongly agree). The latter was selected as it is more suitable in evaluating degrees of difference rather than employing dichotomous variables to indicate the presence or absence of a particular practice. Other variables related to the questions in the study aim to define the company profiles. Each question had its own specific scale designed for its subject, producing the following variables: type of generic strategy, degree of ICT use, software used, the existence of a role in managemen<sup>t</sup> control, types of control tools employed, questions pertaining to culture and organizational structure, number of employees, range of annual sales, sector, among others (see Appendix A.1).

The survey application method is multichannel (personal, telephone, and online). It was applied to company managers throughout the months of September and October of 2018. The process consisted of sending an email invitation to all the mentioned databases, specifically to 1200 SMEs at the national level. The objectives of the study were explained, and the participants were informed by email regarding the structured survey addressed to company managers. In the second stage of the process, an access link to the survey was sent, followed by telephone calls and corporate visits, with the aim of improving the survey response rate.

#### *3.5. Variables Measurement*

Management control variables [10] and performance [11,40–42] are measured through linguistic sentences, as proposed in the study. Given the lack of empirical research with the proposed verbal sentences to measure what the study requires, each dimension has been coupled with several authors (see Appendix A.2).

The scale employed in the survey is the Likert scale: strongly agree, agree, neither agree nor disagree, disagree, and strongly disagree. To proceed with fuzzification and defuzzification, the output tags are high improvement, medium improvement, and no improvement, respectively (see Table 2).

Appendix A.1 shows the fuzzy inference input variables. There are 25 variables that represent "Management control" grouped in: Beliefs, Interactive Control, Diagnostics, and Performance. Following [11] works and seven variables to measure "Performance" [11,40–42], all variables are qualitative. Their evaluation depends on the perception of experts. Given their nature and characteristics, fuzzy analysis is the most suitable means to measure them [43].

Thus, by evaluating the control managemen<sup>t</sup> levers used in SMEs, the degree of truth for each linguistic sentence is measured within the [0, 1] range. This is conducted through a diagnostic instrument applied to company directors.
