Ω = {Supplier 1, Supplier 2, Supplier 3, Supplier 4 and Supplier 5}.

According to the experience and the expertise of the acquisition, logistic, and quality areas, the three attributes are evaluated according to Table 1. The evaluations are shown in Table 2.



Following the proposed model (Figure 5) and considering Table 1, fuzzy sets with interval-valued membership functions for each agent are built (see Table 3).

**Table 3.** Fuzzy sets for suppliers (agents).


It must be considered that the ideal agent is built according to the independent auditor's professional judgement, skepticism, and experience. It complies with each category in the best way; therefore, the ideal agent (provider) complies with the attributes in Table 4, according to the acquisitions, quality, and logistics areas. To do so, the ideal provider must have a price degree not higher than 83% and a quality of at least 49.8%. It must also meet the delivery time at least 49.8% of the times.

**Table 4.** Fuzzy sets for suppliers (agents).


Then, we use the addition competency index, Equation (2), between fuzzy sets *Aϕ* <sup>1</sup> , *<sup>A</sup><sup>ϕ</sup>* <sup>2</sup> , *<sup>A</sup><sup>ϕ</sup>* <sup>3</sup> , *<sup>A</sup><sup>ϕ</sup>* <sup>4</sup> , *<sup>A</sup><sup>ϕ</sup>* <sup>5</sup> , and the ideal set *<sup>I</sup>*. To calculate *<sup>μ</sup><sup>I</sup><sup>ϕ</sup>* (*<sup>A</sup><sup>ϕ</sup>* <sup>1</sup> ), the data in Table 3 and Equation (2). Then,

$$
\mu\_{\overline{I}^{\mathfrak{p}}}(\widetilde{A}\_1^{\mathfrak{p}}) = \frac{1}{3} \sum\_{i=1}^n \mu\_{\overline{I}^{\mathfrak{p}}}^{x\_i}(\widetilde{A}\_1^{\mathfrak{p}}),
$$

where

$$\begin{split} \mu^{x\_1}\_{I^\varphi}(\tilde{A}^\theta\_1) &= \frac{\log([33.2, 66.4] \cap [49.8, 83])}{\log([33.2, 66.4] \cup [49.8, 83])} = \frac{\log([49.8, 66.4])}{\log([33.2, 83])} = 0.33, \\ \mu^{x\_2}\_{I^\varphi}(\tilde{A}^\rho\_1) &= \frac{\log([33.2, 66.4] \cap [49.8, 83])}{\log([33.2, 66.4] \cup [49.8, 83])} = \frac{\log([49.8, 66.4])}{\log([33.2, 83])} = 0.33, \\ \mu^3\_{I^\varphi}(\tilde{A}^\rho\_1) &= \frac{\log([0.33.2] \cap [49.8, 83])}{\log([0.33.2] \cup [49.8, 83])} = \frac{\log(\phi)}{\log([33.2, 83])} = 0. \end{split}$$

Figure 6 illustrates the intersection and union of sets for membership functions *μx*1 *<sup>I</sup><sup>ϕ</sup>* , *<sup>μ</sup>x*<sup>2</sup> *Iϕ* .

**Figure 6.** The unions and intersections of the supplier evaluation.

The model seeks to apply an addition competency index to measure the similarity of the agents under evaluation. Thus,

$$
\mu\_{\bar{I}^\varphi}(\tilde{A}\_1^\varphi) = \frac{1}{3}(0.33 + 0.33 + 0) = 0.22.
$$

In an analogous way, the adequacy index for the rest of the agents is calculated. The results are summarized in Table 5.

**Table 5.** Adequacy index.


Clearly, agents *A*<sup>2</sup> and *A*<sup>5</sup> show a higher adequacy index and, in consequence, they (providers) comply with the internal control process for raw material in the best way, according to the acquisitions policies of the company. That is, both agents exert a good control environment, a governance aligned with the study, and an evaluation of internal control by the independent auditor.

Adequate segregation of the functions regarding purchase authorization and price quote when acquiring goods is key to evaluate internal control compliance. Then, product requirements must be followed according to the stock level indicated by the management. In addition, product reception at the warehouse must be observed.

The area manager must check that the process complies with the product quality attributes (weight, shape, smell, color, and packing, among others) for storage at the warehouse. The agent desirably implements an efficient control of the purchase orders and the pre-numbered reception notes related to every acquisition. They should also take frequent physical inventory counts. Both processes include the verification of legal-fiscal documentation, invoice review, prices and calculations, and a review against internal documents before accounting record and the creation of a liability (payment). Then, the manager's authorization degree and hierarchical level to contract and guarantee liabilities are reviewed.

Finally, logistic internal controls are fundamental to performing transactions, product sales by areas or online or any others determined by the management. In the end, the product is delivered to its destination efficiently. The authorization and sales documentation, along with lists of clients, prices, discounts, returns, and bonuses are also necessary to the right logistics.

To comply with the processes, the staff member in charge must authorize the appropriate segregation of clients' orders, payments (cash or credit), shipment, invoicing, credit notes, delivery dates and routes, physical custody, insurance, finance systems, and collaterals or pledges. Then the records are entered in the accounts.

The auditor evaluates the policies and internal control processes of the entity based on their professional assessment and experience, skepticism, and fuzzy governance model.

#### **5. Discussion**

The internal control system and the application of policies and procedures have long been studied by external auditors since they are enforceable in the professional-legal practice [21] within the framework of international audit standards [4]. Furthermore, the presentation of a financial notice is preceded by the study and evaluation of internal control [5]. This obligation is supported by the legislation [11] of international governments aiming at protecting investors and reducing financial disasters [1,2]. After the compliance with the code of best corporate practices [8], the governance system evaluates [9,22] its internal control system involved in accounting laws and standards [11] only from a qualitative perspective. The evaluation lacks mathematical analyses [39] without having a scientific confront.

The authors discuss the relevance of internal control as a cornerstone of governance [16], and good corporate practices [23] without alluding to a mathematical model. In our analysis, we state that mathematical models are binding to social sciences. It is, therefore, a novel way to relate laws [13,14] and audit (accounting) standards with fuzzy sets (fuzzy logic). Results show that control policies and procedures for the acquisition of products (pricing, quality, and delivery time ) indicate a good governance management [12] since the company guarantees product supply to its clients. In addition, the fact of comparing a group of providers opens the door to different possibilities to acquire the product; that is, under better economic, resource, and speed conditions. These attributes promote the decision making of the management [25], in turn considering the rest of the providers as second or third sources of raw material.

Fuzzy logic as an agent of the governance system promotes decision making for product acquisition, leading to constant inventory rotation. The latter, in turn, facilitates corporate finances by opening market opportunities and thus charging clients due to sales increase. Liquidity increases and liabilities decrease, so investing in other assets, as financial instruments or companies, is possible. This is one of the benefits of fuzzy logic as a governance system.

The limitation of this study is its focus on the warehouse and purchasing areas of the corporation. Although we know that it is substantive, inventory control should be studied from a corporate system standpoint to relate it with internal control policies and procedures linked to other areas as sales and finance. This would complete the operation flow and allow a thorough understanding as to how supervising the application of policies and processes, or the lack thereof, increases corporate risk [38]. These research lines remain open for an integral study.

The fuzzy logic method is used since it is adequate for the study of attributes that can hardly be mathematically expressed. Its great potential lies in the possibility to express operations through everyday-use words. It is a tool used in administrative areas, as accounting and finance, of companies [7,38]. The fuzzy governance model is presented [30] as a management system for internal control policies and procedures, focusing on a legal framework to fulfill economic activities, necessary for decision making in corporate governance.

#### **6. Conclusions**

This work shows the inductive-deductive interaction between the implementation of fuzzy models and the qualitative attributes defined in the internal law of organizations. This document proposes a tool to evaluate the governance degree of agents belonging to corporate governance as a dynamic complex system.

Processes, including those non-written, are frequently absent and there is no documentary information to carry out the study and compliance evaluation of policies and internal control procedures within organizations. Fuzzy logic is a novel and efficient tool to study the compliance level of processes, especially that of those non-written.

The auditor observes process compliance and thus moves forward in the analysis of the study and evaluation of the internal control. All of this is based on the auditor's professional experience in the stock market, technical control, professional judgement, discretion, and skepticism exhibited during the examination of the financial information. The auditor uses a fuzzy model to identify the abilities and skills of other professionals, and materializes the technical control, professional judgement, skepticism, and professional experience. These qualities are hardly quantifiable using the known techniques.

Statutory commercial and securities law of accounting and auditing standards and agents of corporate governance concatenate into a complex system. Then, governance is exerted through inter- and transdisciplinary interactions involved in an established dynamic between areas and hierarchical levels of the organization.

**Author Contributions:** E.M.-R. and C.L. conceived the presented idea and they show how mathematics contains methodologies, such as fuzzy logic, which are useful tools for accounting and administrative sciences to evaluate qualitative attributes that are difficult to measure. E.M.-R. contributed to the acquisition, analysis, and interpretation of data for the work. C.L. developed the theory and performed the computations of fuzzy set theory. E.T.-E. searched the existing literature and data collection. E.M.-R., C.L. and E.T.-E. contributed to the analysis of the results and to the writing of the manuscript. All authors discussed the results and contributed to the final manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** The APC was funded by the Universidad la Salle Mexico, project SAD-30/20.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** This research was funded by Universidad La Salle México and is part of a research line and application of accounting and auditing of companies.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**

