Evidence shows that elsewhere in most developed and developing economies, SMEs employ the largest proportion of the workforce. In Brazil, these businesses account for approximately eighteen million formal companies, employing most of the workforce, from agricultural products to the cultural sector (Barbosa et al., 2022). Besides, Brazil had 8,863 exporting SMEs in 2017, which represented 40.8% of the country’s exporting companies in the year, with 17.8% referring to micro-enterprises and 23.1% to small businesses (Serviço de Apoio às Micro e Pequenas Empresas Brasileiras [SEBRAE], 2022).

Some studies evaluating the impacts of COVID-19 for SMEs were carried out for the Brazilian context (Bretas & Alon, 2020; Dweck, 2020; Pereira & Patel, 2022; Rediske et al., 2022; Reis et al., 2021), that operate in different sectors; with a focus on commerce and services (Marques et al., 2021), on educational institutions (A. D. S. M. Costa et al., 2020; B. G. S. Costa et al., 2022; Dias & Ramos, 2022), on the strategies used (Wecker et al., 2020). Thus far, however, it has not been made clear how SME business-related variables impact on the perceived utility of business-criteria, particularly during the COVID-19 pandemic. Precisely, this research focused on five major business-criteria to capture the managerial perceptions of the pandemic impact to SME performance: business impact (whether it remained without operational changes or was affect either by temporary or permanently closure) (Bartik et al., 2020; Latham, 2009); business operation (whether its economic activity level increased, decreased or remained stable) (Dess & Robinson, 1984; S. Singh et al., 2016) ; employee dismissal (whether its jobs were spared during the pandemic or not); loan success (whether it could borrow working capital from banks to sustain their operation or not); and crisis duration (how long pandemic impacts were perceived to last, despite lockdowns, governmental support etc.) (Brown et al., 2020; Deyoung et al., 2015). On the other hand, a comprehensive number of business-related variables, encompassing socio-demographic issues both related to the SMEs and the respondents themselves were targeted as possible perception drivers. As regards the respondents, their academic level, and their respective age; as regards the SMEs, their relative size, time in business, business type, economic sector, and the respective Brazilian State where they are located (Lim et al., 2020; Schepers et al., 2021).

The distinctive methodological approach offered by this research is twofold. First, by unveiling, through the transfer entropy approach, the cause-effect and feedback relationships among major business-criteria, based on the distributional profile of the respondents’ perceptions. Information entropy is a well-stablished concept related to the reliability of a dataset (Núñez et al., 1996). The maximal entropy principle states that the probability distribution which best represents the current state of knowledge for a given business-criteria is the one with largest entropy (Peter et al., 2010) . Second, and differently from previous research, this paper aims at answering how socio-demographic, business-related, variables impact on the perceived utility of distinct business-criteria in Brazilian SME. By computing the information entropy of the distribution of perceptions for each criterion, it becomes possible to focus on the most meaningful criteria for policy making, and their socio-demographic drivers, for which it is not possible to be ascertained a priori.

The impact of the COVID-19 pandemic on SMEs has led to reflection and attention on the SME ecosystem and has attracted the awareness of academics and practitioners (Bretas & Alon, 2020; Cepel et al., 2020; A. D. S. M. Costa et al., 2020; B. G. S. Costa et al., 2022; Habachi & Haddad, 2021; Kamaldeep, 2021; Pereira & Patel, 2022; Reis et al., 2021;). Most of this literature on COVID-19 and SMEs reveals an understanding of how companies have responded to or been impacted by the effects of COVID-19 (Bretas & Alon, 2020; A. D. S. M. Costa et al., 2020; Dejardin et al., 2023; Dweck, 2020; Habachi & Haddad, 2021; Kamaldeep, 2021; Ma et al., 2021; Pereira & Patel, 2022; Rediske et al., 2022; Reis et al., 2021). In other words, these studies tend to describe the dynamics of COVID-19 and its effects on SMEs, mostly based on descriptive studies. Although the body of research has generated relevant results on the subject, the success, or difficulties of SMEs during the pandemic has not yet been understood, with undeveloped aspects regarding the effect of the pandemic on SMEs from emerging countries.

Spurred on by the research gaps, this original study reports on a series of survey data collected in Brazilian SME by means of a novel neural-MCDM (multi-criteria decision making) model structured in three stages (Sheng-Hshiung et al., 1997; T. C. Wang & Lee, 2009). This model is proven capable of deriving unbiased utility functions for distinct business-criteria based on information entropy levels captured from the respective respondent´s perceptions. In fact, information entropy is the cornerstone method used in this research to assess the perceived importance of each business-criteria, based on weights computed using the recent SWARA model. Compared with other methods, information entropy provides the benefits of lower bias and higher robustness to unconsidered assumptions, which can lead to a more comprehensive interpretation of the results as regards how the utility of distinct attributes, as derived by COPRAS (Zavadskas & Kaklauskas, 1996), are perceived by distinct demographic groups. Results indicate that analyzing each criterion in an isolated fashion, crisis duration, business operation and employee dismissal appears as the most relevant criterions, as expected due to the economic moment caused by the pandemic. The two least relevant criteria, loan success and business impact, relate to actions that could be taken to keep SMEs running even despite lockdown interruptions. Most SMEs suffered from business interruptions that may have caused operational changes, thus yielding lower economic activity. Besides, while most of them did not find financial support in banking loans for working capital, they are so diminished in size (self-entrepreneurs) that employee dismissal presented a limited impact on explaining the lower utility function levels. In general, our paper contributes to our understanding of the impact of COVID-19 on the small business ecosystem in Brazil. This survey on the impact of the COVID-19 pandemic on the operations of small and medium-sized enterprises (SMEs) in Brazil is, to date, the most comprehensive and representative in the sector. The study involved 7,000 companies from different segments and regions. The rest of this paper includes four sections. Literature review is presented in Section 2, while methodology is presented in Section 3. Section 4 focuses on the analysis and discussion of results, and conclusions are elaborated in Section 5.

The SWARA steps used in this research are duly described next (Stanujkic et al., 2015).

Step 1: Sort business-criteria from the highest to the lowest based on the information entropy ranking for each criterion.

S tep 2: Assign the null value for the preference of the first business-criteria. Allocate preferences to the second most important business-criteria; repeat this step until the least important business-criteria is reached. These preferences are computed by comparing a given business-criteria with the first one with highest entropy. Compute their pairwise relative importance, denoted by S _j , which shows the ratio of this comparison.

Step 3: Set-up pairwise efficiency criteria by K _j :

K j = 1 , j = 1 S j + 1 , j > 1 (8)

Step 4: Compute relative (q _j )weights ( based on sorted pairwise efficiency with respect to the importance criterion rank:

q j = 1                   j = 1 K j - 1 K j                   j > 1 (9)

Step 5: Compute final weights as W j = q j ∑ k = 1 n q k , where W _j denotes the weight of each criterion j.

COPRAS was first introduced more than two decades ago by Zavadskas and Kaklauskas (1996). Since then, several different researches have been published on possible alternative ways for combining SWARA and COPRAS (Zolfani & Bahrami, 2014; Nakhaei et al., 2016; Valipour et al., 2017); SWARA and Fuzzy COPRAS (Bekar et al., 2016; Yazdani et al., 2011); and COPRAS and other MCDMs (Aghdaie et al., 2012; Ecer, 2014; Fouladgar et al., 2012; Rezaeiniya et al., 2012; Zolfani et al., 2012). The next lines briefly present the major steps of the COPRAS method applied in this research for deriving utility functions based on different business-criteria importance weights (cf. previous section):

Step 1: Create a decision-making matrix X, containing m respondents and n business-criteria:

X = a 11 … a 1 n ⋮ ⋱ ⋮ a m 1 ⋯ a m n   i = 1,2 , . . . n ; j = 1,2 , . . . , m (10)

Step 2: Normalize the decision matrix X by computing:

x i j ¯ = x i j ∑ j = 1 n x i j (11)

Then the decision matrix will be:

X ¯ = x ¯ 11 … x ¯ 1 n ⋮ ⋱ ⋮ x ¯ m 1 ⋯ x ¯ m n (12)

Step 3: Compute the weighted normalized decision matrix by means of:

x i j ^ = x ¯ i j × w i j ; i = 1,2 , . . . , n ; j = 1,2 , . . . , m (13)

Therefore,

X ^ = x ^ 11 … x ^ 1 n ⋮ ⋱ ⋮ x ^ m 1 ⋯ x ^ m n ; i = 1,2 , . . . , n ; j = 1,2 , . . . , m (14)

Step 4: Sum-up the larger values that are more preferable, named as P ⁱ :

P i = ∑ j = 1 k x i j ¯ (15)

Step 5: Sum-up the smaller values that are more preferable, named as R _i :

R i = ∑ j = k + 1 k x i j ¯ (16)

Then the number of business-criteria that should be minimized is given by the difference m-k.

Step 6: Minimize R _i observing eq. (8):

R m i n = m i n i R i ;   i = 1,2 , … ,   n (17)

Step 7: Compute the relative significance of each business-criterion as given ǫ_i:

Q i = P i + R m i n ∑ i = 1 n R i R i ∑ i = 1 n R m i n R i (18)

Step 8: Identify the optimal business-criterion i, given by K, as is illustrated:

K = m a x i Q i ; i = 1,2 , . . . , n (19)

Step 9: Prioritize business-criteria in a descending order.

Step 10: Determine the utility degree N of each subsequent business-criterion i, given as:

N i = Q i Q m a x (20)

Density plots for the business-criteria weights computed using SWARA are depicted in Figure 1, based on the information entropy distributions provided by respondents for each criterion. Analyzing each criterion in an isolated fashion, crisis duration appears as the most relevant criterion, followed by business operation and employee dismissal as expected due to the singular economic moment caused by the pandemic. These three most relevant criteria indicate that SME concerns are mostly related to lockdown decisions and the consequent impact on economic activity and employment level. The two least relevant criteria, loan success and business impact, relate to actions that could be taken to keep SMEs running even despite lockdown interruptions. Besides, this importance imbalance among business-criteria is also reflected on the overall utility function distribution: SMEs tend perceive such utility as low - most utility function values are below 0.50 - what in some sort anticipates the nature of problem faced during the pandemic in light of the response levels/factors for each business-criteria as depicted in Table 2. Most SMEs suffered from business interruptions that may have caused operational changes, thus yielding lower economic activity. Besides, while most of them did not find financial support in banking loans for working capital, they are so diminished in size (self-entrepreneurs) that employee dismissal presented a limited impact on explaining the lower utility function levels.

Figure 1 Source: Elaborated by the authors.

Figure 2 Source: Elaborated by the authors.

Table 2

Business criteria*	Frequency distribution for each response/ factor level (number in brackets denote the response level)
Business Impact (+)	Permanently Closed Business (1)	0.32%	Temporary Closed Business (2)	30.22%	Business with Operational changes (3)	56.93%	Business without Operational changes (4)	12.53%
Business Operation (+)	Lesser (1)	85.63%	Equal (2)	7.90%	Greater (3)	6.47%
Employee Dismissal (-)	Without Dismissal (1)	39.85%	Business without employees (2)	48.48%	With Dismissal (3)	11.67%
Loan Sucess (+)	Loan Denied (1)	79.03%	Waiting Response (2)	9.89%	Loan Approved (3)	11.08%
Crisis Duration(-)	Descriptives
	Min	0	Max	365	Mean	11.89	SD	11.64

* Signs are related to the positive or negative impact of a given criteria on overall utility. They reflect the factor/response levels for each criterion, observing intrinsic relations such as “the higher the better,” “the higher the worse” w.r.t. utility function values.

Transfer entropy and neural network results for cause-effect relationships among business-criteria and business-related variables in Brazilian SME are depicted in Figure 3. Table 3 also reports on the best ANN architecture found for each regression, after cross-validating the originally trained models with a randomly selected 20% of the sample. One may easily note that business operation is the most critical criteria: it impacts three other criteria (crisis duration, employee dismissal and business impact), and it is only impacted by one (loan success). Greater economic activity not only impacts on the respondent´s perceptions about the duration of lockdowns and the persistence of pandemic effects but can also revert decisions with respect to reduction in workforce or even shutting down the business. Loan success is the second most impact criterion, deeply impacting the continuity of economic activity levels; it could be considered a pure exogenous criterion since it is not impacted by any other business-criteria. Consistent with Deyoung et al. (2015) and Maffioli et al. (2017), the availability of credit resources for SMEs directly impacts the business operation.

On the other hand, business impact is purely endogenous, its perception is the resultant of the countervailing forces represented by economic activity level; reduction of labor force; and successful working capital loans for sustaining business during the pandemic. These pure exogenous and endogenous business criteria may explain why their perceived utility is high (COPRAS function present a positive impact highlighted in green). Hence, more resilient SMEs - without operational changes - are those with 5-10 years’ operating in the services and construction sectors. On the other hand, SMEs that suffered the most with lockdowns are those related to food and technology industries. As regards banking support, food services SMEs were more successful in borrowing working capital from banks than novel SMEs that operate in the gym, pet shop, and educational services in general. It is important to note that, regardless of the business-criteria, the educational respondent profile, and the state of location of the SME were also found to be pretty heterogeneous, results that suggest that perceptions and the utility functions on the distinct business-criteria still depends on whether the SME is located in poorer or richer Brazilian states or on whether the self-entrepreneurs are illiterate or not. This is crucial evidence of the impact of formal education on the survival of SMEs during the COVID-19 pandemic crisis. The absence of adequate education to run a business can make it difficult to deploy dynamic capabilities, or technological capabilities. This is an underlying and relevant variable in any business model, in any sector. Education is important for small and medium-sized enterprises (SMEs) because it can help to improve productivity, increase competitiveness, and create new jobs. Differences in local financial development particularly affect corporate finance decisions of small and medium-sized firms (Fasano & Deloof, 2021). The full set of ANN results are depicted in the Appendix.

Table 3

Business-Criteria	Layers	Neurons per Layer	L1 Regularization	L2 Regularization	Error Measure	Validation Error
Business Impact	1	35	1.00E-07	1.00E-07	ACC	71.22%
Business Operation	2	5	1.00E-05	1.00E-07	ACC	76.37%
Employee Dismissal	4	30	1.00E-05	1.00E-05	ACC	70.08%
Loan Sucess	3	35	1.00E-06	1.00E-05	ACC	53.91%
Crisis Duration	3	5	1.00E-06	1.00E-05	MAE	0.53

Validation with 20% of total dataset observations. MAE stands for mean absolute error, while ACC stands for accuracy, that is, the fraction of correct predictions. Readers should note that for the first four business-criteria, a classification neural network model was performed to regress socio-demographic variables onto a respective response level.

Figure 3 Key: List of the five most relevant positive in green and negative in red. Note: All results were controlled by COPRAS utility function scores. Source: Elaborated by the authors.

The study aimed to propose a novel evaluation model for addressing the impact of COVID-19 on SMES through managerial perceptions. A novel entropy-weighted utility function approach is proposed here, followed by artificial neural network regression to map which SME business-related variables drivers the most the perceived utility of each SME business-criteria during the pandemic. Neural network regressions were used to explain the managerial perceptions on each business criterion during the pandemic considering each business variable, while controlling for the respective criterion utility.

The entropy-weighted utility function approach and the ANN regression were impactful in figuring out the SMEs business-related variables that most drive the perceived utility of each SME business criterion during the pandemic for some reasons: 1) it considers the uncertainty and variability of data by incorporating entropy calculations. This helps in managing the complexity of business-related variables and their impact on perceived utility. This type of approach could be used to any unpredictable and rapidly changing situations; 2) by considering the weights (Olden & Jackson, 2002; Olden et al., 2004), decision-makers can prioritize and focus on the variables that have the greatest impact on business outcomes; 3) By using ANN regression, the business-related variables and their influence on the perceived utility can be mapped in a non-linear manner. But while the proposed model offers valuable insights, there are certain limitations, such as: it requires a precise specification of the utility function and the probability distribution of the outcomes, which may not be easy to elicit in real-world problems.

Forthcoming studies might conduct more research on these managerial perceptions issues to improve the proposed model, additional granular examinations of isolated business type, or focus on the smallest SMEs (e.g., less than 5 employees). Fasano and Deloof (2021) identified that the distribution of financial credit, with the purpose of lengthening payment terms, to the supply chain of a given SME can be more effective than the resource directly allocated in the company, depending on the context and SME’s operating sector. This was not investigated in this study and, if studied, this aspect might contribute to the design of policies for SMEs. SMEs’ financial performance could be investigated considering business impact and operational functions, including SME structure and owner capability. The role of education in building dynamic strategies and technological capacity is critical for SMEs, especially in times of crisis. There is a gap on this subject in the literature on strategy and business resilience for SMEs.

Finally, the model proposed in this article enables capturing intricate relationships that may not be easily identifiable through traditional statistical methods. By understanding the described transformations in steps for SWARA and COPRAS, and how the ANN was applied, researchers can assess whether this method is suitable for their research problem.