**1. Introduction**

On the one hand, according to the fifth longitudinal survey of companies of the Chilean Ministry of Economy, small companies are extremely important for the national economy, as they represent 52.5% of all companies and employ 38.7% of all workers. These companies have many problems, often lacking the time, resources, or necessary information to deal with organizational performance [1].

On the other hand, managemen<sup>t</sup> control is the process that managers use to aid the decision-making of the members within an organization. This eases the application and alignment of chosen strategies in the organization, thus achieving the pre-established objectives and benefiting the overall performance of the company [2–4].

Notwithstanding, the international literature recognizes that managemen<sup>t</sup> control tools are fundamental for the efficient and effective managemen<sup>t</sup> of any business [5], while information and planning systems are useful tools to obtain corporate strategic objectives [6]. When they are met, the goals and reason of existence of companies are achieved, making them an important aspect of improving organizational performance [4,5,7,8].

Besides, evidence shows that the level of use of financial managemen<sup>t</sup> tools directly affects performance, with the most common ones (according to literature, and interpreted as levers) being budget, long-term planning, support systems for decision making, and financial and non-financial performance [7,8]. There is limited literature regarding the existence of a holistic system that aggregates all these tools, which would allow for the careful study on how each lever (pack of tools) affects the organizational performance of SMEs (Small and Medium-sized companies). In addition, there are limited empirical studies based on scientific data that examine how financial managemen<sup>t</sup> practices affect SMEs [9]. Nonetheless, existing literature shows a positive outcome of the use of managemen<sup>t</sup> control tools in SMEs to maximize opportunities, operational efficiency, profit, reliability of administrative information, and finances [10].

**Citation:** Nicolas, C.; Müller, J.; Arroyo-Cañada, F.-J. A Fuzzy Inference System for Management Control Tools. *Mathematics* **2021**, *9*, 2145. https://doi.org/10.3390/ math9172145

Academic Editor: Ioan Dzitac

Received: 4 August 2021 Accepted: 29 August 2021 Published: 2 September 2021

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The purpose of this investigation is to explore the influence of the use of managemen<sup>t</sup> control tools on the organizational performance of SMEs. According to the latter, the question of this research is: How do we apply an MFIS in Management Control Tools? What is the degree of use of each control lever in Chilean SMEs?

The present research explores, for the first time, the use of four levers of managemen<sup>t</sup> control tools in SMEs (belief systems, boundary systems, interactive control systems, and diagnostic control systems) [11]. The individual influence of each of these tools on performance make up the four hypotheses of the research:

**Hypothesis 1 (H1).** *The beliefs lever is present as a management control tool in SMEs.*

**Hypothesis 2 (H2).** *The boundary lever is present as a management control tool in SMEs.*

**Hypothesis 3 (H3).** *The diagnostics control lever is present as a management control tool in SMEs.*

**Hypothesis 4 (H4).** *The interactive control lever is present as a management control tool in SMEs.*

The methodological approach taken in this study is a mixed methodology based on factorial analysis and Mamdani fuzzy inference system (MFIS). This fuzzy model is built around the MATLAB Software and its Fuzzy Logic Toolbox. The investigation is empirically applied in Chile, and the proposed fuzzy model is used to evaluate the degree of presence of these levers in SMEs, as well as their relationship with the levels of financial and non-financial performance.

Fuzzy inference systems (FIS) are methodologies that express knowledge and inaccurate data; in Web of Science, there are 1,314 records on the topic. FIS are studied mainly in computer science artificial intelligence (27%), electrical electronic engineering (20%), interdisciplinary computer science applications (10%), and others ( 43%). The authors with the highest number of publications are P. Melin, O. Castillo, and O. Kisi (Web of Science).

Mamdani fuzzy inference system (MFIS) shows 92 records on the topic; the main areas studied are computer science artificial intelligence (25%), electrical electronic engineering (15%), computer science theory methods (12%), environmental sciences (12%), and applied mathematics (12%). The authors with the highest number of publications in mathematics categories are B. Jayaram and M. Stepnicka (Web of Science).

Unlike other studies that apply MFIS in management, this paper proposes a diagnostic model of the four levers of managemen<sup>t</sup> control in a holistic view; other studies are focused on customer requests or specific tools, such as the Balanced Scorecard (BSC) [12].

The importance and originality of this study are that it explores and applies the fuzzy inference system with linguistic control rules to measure the use of the managemen<sup>t</sup> control tools in SMEs and their relationship with financial and non-financial performance. Mamdani's fuzzy inference system is advantageous for this study since it is intuitive, well-suited to human inputs, more interpretable and rule-based, and has widespread acceptance [13].

This study contributes to managemen<sup>t</sup> control field and SMEs, which are companies that have survived the start-up stage and they are currently in the positioning and growth stages [14]. In addition, as established at the beginning, these companies are important for the national economy since they provide many jobs. From the area of managemen<sup>t</sup> control, we can help them to continue growing by supporting them in understanding the importance of applying certain managemen<sup>t</sup> control tools. Furthermore, this type of research helps other SMEs incorporate managemen<sup>t</sup> tools in order to improve their organizational performance. It must also be noted that similar studies elaborated in Latin America as a whole are scarce.

The main issues addressed in this paper are: Section 1 of this paper will review the literature on managemen<sup>t</sup> control and managemen<sup>t</sup> control research in SMEs with fuzzy logic; Section 2 gives a brief overview of the recent methodology; Section 3 presents the results of the research, focusing on the three key themes that are: Descriptive Analysis, Measurement Scale, Fuzzy Inference Systems and Parameterization of Membership Functions, Summary of Fuzzy Indicators; Finally, the Discussion and Conclusions are presented.

## **2. Literature Review**

Various studies identify the structural weaknesses of SMEs regarding their lack in the use of tools to create strategies that can manage projects in the medium and long term [6,7,9,10]. In addition, their methods in managemen<sup>t</sup> control, administration, finances, accounting, and operations are done in an informal and intuitive manner, disregarding the use of decision-making tools. This is because many of them are created by the experience of staff who have worked in other companies, thus bringing with them technical but not administrative knowledge [15]. In retrospect, SMEs usually lack time, resources, or necessary information (or the skill set required to collect and evaluate this information) to measure organizational performance [1].

However, SMEs possess crucial competitive advantages: (1) Their size allows them to respond rapidly to changes in their environment, easing their integration into the productive chain, (2) They tend to be efficient providers of intermediate or final goods and services. They do, however, have the following disadvantages: (1) They are vulnerable to recessive cycles and slow economic growth, (2) They cannot surpass technical and non-technical barriers of market entry on their own, (3) They are unable to develop barriers to protect their income in specific market segments and niches [16].
