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go, 2021).
In this research, the combination of fuzzy logic with ZBB specifies, among its advantages (Báez and Bravo, 2020): 1) formalization and simulation of expert reports in the conduction and standardization of a process; 2) provides simple answers to difficult modeling procedures; 3) takes into consideration several variables and their weighted fusion determines the magnitude of the influence; 4) continuously considers cases or exceptions of different nature, integrating them in the solution; 5)
allows the implementation of multi-criteria strategies incorporating the knowledge of experts.
Specifically, the central hypothesis of the research is outlined as follows:
H0: Zero-Based Budgeting, supported by fuzzy logic, allows SMEs to make optimal and accurate decisions, based on previously defined thresholds, taking advantage of available resources efficiently in restrictive contexts.
In Zero-Based Budgeting, each expense (and income stream) must be explained and justified annually, and in its entirety to senior management (Messer, 2020). In this type of budget, substantive adjustments are made to expenditures (Tovar, 2015), and it is assumed to be one of the most important tools for business and social development (Tacuba, 2016), as it allows concentrating productive programs that address specific problems.
The ZBB: 1) is a methodology that provides detailed information on the resources required to achieve the desired objectives, 2) outlines concrete goals based on a company's priorities and, 3) defines to which items these resources will be directed. By performing a cost-benefit analysis to avoid duplication of functions and programs, based on the design of decision packages, a function or a specific activity can be described, which helps to evaluate and rank it in comparison with other activities.
These tools allow reevaluating programs and expenditures always starting from zero, evaluating and justifying the amount and need of each area and each program when the agencies are spending more than necessary, and incorporating the evaluation in the preparation stage and not only in the results stage (Tacuba, 2016). In other words, it is managed under the principle of administration or management by objectives, leading efforts towards the achievement of goals, directing resources with greater certainty towards their realization.
H1: Zero-Based Budgeting, based on fuzzy logic, eliminates inertial and discretionary budget allocation criteria.
The ZBB follows an accounting approach oriented to operational efficiency and business redirection through strategic management, is accompanied by long-term savings, cost optimization and investment in innovation (Tacuba and Chávez, 2018), the latter essential in difficult and challenging times when revenues and growth are reduced (Madrid-Guijarro et al., 2016).
H2: Zero-Based Budgeting, supported by fuzzy logic, allows the allocation of resources to SME objectives by prioritizing them;
The ZBB is built based on programs, and their impact on revenues and problem solving (Huerta, 2016; Brotons and Sansalvador, 2015), however in contexts of high uncertainty such as the current ones (pandemic resulting from COVID-19) the situation of institutions changes, even more so that of small and medium-sized enterprises.
H3: Zero-Based Budgeting, supported by fuzzy logic, allows for periodic re-evaluation of objectives based on predetermined thresholds for decision-making.
The combination of ZBB with fuzzy logic projects decision thresholds from the highest certainty, optimizing resources and directing objectives towards what is essential in companies, particularly SMEs. Discretionally is avoided, and resources are projected towards what is really important and crucial in the companies.
In the organizational context, companies are subjective units, which perform in dissonant, imprecise, vague, ambiguous, inaccurate and uncertain or probabilistic realities by nature (Restrepo and Vanegas, 2015). Fuzzy logic is projected to handle the present uncertainty, based on the nonlinear correspondence between one or several input variables and an output variable, facilitating a basis from which decisions can be made or patterns can be defined that are represented by non-exact values (Flores and García, 2013). It is precisely these situations where the application of fuzzy mathematics (triangular fuzzy numbers and fuzzy subsets) is proposed as a way of dealing with the uncertainty prevailing in the context.
Under a theoretical-hypothetical relationship, the following model for the research is specified as it is shown in Diagram I.
3. Methodology
Fuzzy numbers as a finite or infinite sequence of confidence intervals represent one of the greatest contributions to the knowledge of fuzzy logic, being of great support and benefit for the development of knowledge (Kaufmann and Gil-Aluja, 1987). From this perspective, the theory of fuzzy subsets is applied to business management for the treatment of uncertainty (Kaufmann and Gil-Aluja, 1986), that is, fuzzy logic is incorporated into organizational problems.
Lazzari (1997) explains that an TrFN is determined only by four real numbers (the minimum value, the maximum value and the values of the highest level of assumption); the TrFN will represent the opinion of experts in a wider range. According to Solano (2019), by using tools supported by mathematical models, the assumptions about the relationships between variables and elements of the complex real system could be simplified.
The structure of the ZBB will be based on the application of TpFN, representative values between which a certain event can occur. In this way, the objectives to be achieved in the next period by each department of the company are established, with the purpose of making future predictions based on the TpFN, a fuzzy logic technique. The information is derived from a survey of key personnel in each area of the organization, as well as managers and administrators familiar with the daily management of the company.
Based on the above, the research is inserted in the quantitative plane, it is predictive and projective, and it applies advanced tools offered by fuzzy logic, such as the technique of expertization and TrFN; tools that try to reduce uncertainty with the desired degree of certainty for the future (Luna et al., 2018; Tinto et al., 2016).
4. Results
Within fuzzy logic, one of the most widely used tools of expertizing is the hendecadarian scale used to reduce uncertainty and adjust examined values (Alvarez et al., 2020). Rico and Tinto (2010), propose the use of techniques developed based on the theory of fuzzy subsets, such as expertizing-counter-expertizing, and the theory of forgotten effects in the ex-post treatment of traditional accounting information, with the aim of improving its capacity to support appropriate medium and long-term decision making.
On this basis, Kaufmann and Gil-Aluja (1989), establish that the incidence of one variable with another is expressed by the matrix of forgotten effects, including a greater number of incidences considered as fuzzy elements with a valuation of [0, 1] within an hendecadarian scale, represented at unity as maximum incidence and at zero with no incidence.
The ZBB concept, applying fuzzy logic, expresses the magnitudes corresponding to future periods (Gil-Lafuente et al., 2015). They are data estimated by experts, in one of the most common forms in
the field of uncertainty; TrFN, in addition to this, expose a fuzzy constraint to the overall budget and a criterion for selecting the most appropriate budget for the organization.
The development of a ZBB for a manufacturing company, directs its structure to capture the economic dynamics, optimizing costs to achieve goals in a given period (Luna et al., 2018). Fuzzy logic systems are more flexible and accept the imprecision, subjectivity and vagueness (uncertainty) of the data, allowing to obtain effective solutions to support, in an appropriate way, decision making (Rico and Tinto, 2008).
For the valuation of expert opinion, we resort to the nomenclature introduced by Kaufmann and Gil-Aluja (1989), who state that the introduction of a nuanced valuation between 0 and 1 makes it possible to intervene levels of truth in the notion of incidence: (...) Values from 0 to 1 (the so-called hendecadarian valuation). The principle of Gradual Simultaneity (Kaufmann and Gil-Aluja, 1986) considers that any proposition can be true and false at the same time, as long as one degree is given to truth and one degree to falsehood. Use a real number between 0 and 1, which is simpler and much closer to the skillful way of thinking of man, rehabilitating subjectivism and imprecision (Salazar-Garza, 2012). The hendecadarian scale used for this study is presented in Table I.
Table I. Hendecadarian Scale
PRESUMPTION INCIDENCE
GRADE a
0 Low
0.1 Virtually low
0.2 Very low
0.3 Fairly low
0.4 Lower than high
0.5 As low as high
0.6 Higher than low
0.7 Fairly high
0.8 Very high
0.9 Nearly high
1 High
Source: own elaboration
The first step for the design of the ZBB with fuzzy logic is to outline the objectives to be achieved in the next financial period. As an illustrative example, it is presented for the industrial company "A" in Cuenca (Ecuador); we had the support of key personnel from different areas of the organization, considering the economic and health reality currently faced. Table II shows the objectives to be achieved.
Table II. Objectives to be achieved
№ Objectives
1 Train personnel in the technological field.
2 Increase production margin.
3 Increase market share.
4 Orient marketing processes to new market niches.
5 Incorporate biosafety elements in the work environment.
6 Launch products with new designs.
7 Optimize teleworking.
8 Apply for credit financing from public or private banks.
9 Create innovative management models.
10 Modernize after-sales service.
11 Use social networks for better product positioning.
12 Improve return on investment.
13 Increase customer satisfaction.
14 Optimize costs and expenses.
15 Improve the company's image.
16 Promote job stability.
Source: own elaboration
Once the objectives to be met have been defined, the next step is to develop the technique of fuzzy logic expertise. This tool tries to reduce uncertainty in the information. Luna and Sarmiento (2019) argue that expertise implies the consultation of a defined group of experts in affinity with a given topic, with the intention of limiting uncertainty. For its application in the case under development, the opinion received from 12 officials from different areas of the company is considered, who, according to their opinion, determined the importance of the objectives outlined (Table II), according to the hendecadarian scale (Table I).
As an example, the steps of this technique are presented in relation to the first objective "To train personnel in the technological area". Table III shows the opinion of each of the experts regarding the importance of the first objective. As can be seen in Table III, the responses of 0.6 and 0.7 are repeated twice; 0.8 is repeated five times; 0.9 is repeated twice; and 1 is repeated four times.
Table III. Opinions of expert officials in relation to the objective "To train staff in the technological field".
Respondent 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Response 0.8 1 1 0.9 0.7 0.6 0.8 0.8 0.8 0.9 0.6 1 1 0.8 0.7
Source: own elaboration
Next, frequency normalization is performed; this consists of dividing the frequency values reached in each degree of presumption of the hendecadarian scale by the number of experts (15), thus 2^15 = 0.133; 5^15 = 0.333; and 4^15 = 0.267. The frequencies are accumulated, starting with the summation from the end of the series, until the unit is obtained, from then on, all the values are considered as one (1.00), as shown in Table IV.
Table IV. Normalization and Frequencies Accumulation
DEGREE OF FREQUENCY NORMALIZATION OF ACCUMULATION OF
PRESUMPTION a FREQUENCY FREQUENCIES
0 0 0.000 1.000
0.1 0 0.000 1.000
0.2 0 0.000 1.000
0.3 0 0.000 1.000
0.4 0 0.000 1.000
0.5 0 0.000 1.000
0.6 2/15 0.133 1.000
0.7 2/15 0.133 0.867
0.8 5/15 0.333 0.733
0.9 2/15 0.133 0.400
1 4/15 0.267 0.267
TOTAL 8.267
EXPERTIZED VALUE 0.827
Source: own elaboration
The total obtained in the sum of the accumulation of frequencies is divided by 10, which corresponds to the factors that form the degree of presumption from 0.1 to 1, giving as a result: 8.267 -r 10 = 0.827. This value represents the aggregate opinion of the fifteen experts consulted on the impact of
the objective "To train personnel in the technological field". This technique is developed in a similar way for the other objectives. The results obtained are shown in Table V.
Table V. Threshold determination
№ Objectives Threshold
1 Train personnel in the technological field. 0.827
2 Increase production margin. 0.833
3 Increase market share. 0.593
4 Orient marketing processes to new market niches. 0.813
5 Incorporate biosafety elements in the work environment. 0.760
6 Launch products with new designs. 0.647
7 Optimize teleworking. 0.880
8 Apply for credit financing from public or private banks. 0.793
9 Create innovative management models. 0.593
10 Modernize after-sales service. 0.820
11 Use social networks to better position the product. 0.833
12 Improve return on investment. 0.573
13 Increase customer satisfaction. 0.613
14 Optimize costs and expenses. 0.813
15 Improve the company's image. 0.540
16 Promote job stability. 0.613
Source: own elaboration
In order to define the objectives to be considered in the ZBB, information was gathered from twelve experts in charge of decision making within the company. The opinion provided was again based on the hendecadarian scale, through the application of confidence intervals, represented by a minimum and maximum value (Table VI).
Table VI. Expert opinion "Train staff in the technological field."
EXPERTS RESPONSES
1 [ 0.5 0.9 ]
2 [ 0.6 0.8 ]
3 [ 0.7 0.8 ]
4 [ 0.4 1.0 ]
5 [ 0.5 1.0 ]
6 [ 0.7 1.0 ]
7 [ 0.7 0.7 ]
8 [ 0.5 0.6 ]
9 [ 0.5 0.8 ]
10 [ 0.6 0.9 ]
11 [ 0.4 0.7 ]
12 [ 0.4 0.9 ]
Source: own elaboration
The same procedure of the expert assessment technique explained is followed to find the aggregate opinion of the twelve experts, related to the importance of the first objective (Table VII).
Table VII. Normalization and Frequencies Accumulation
DEGREE OF PRESUMPTION a
FREQUENCY
NORMALIZATION OF FREQUENCY
ACCUMULATION OF FREQUENCIES
0.0 0 0 0.00 0.00 1.00 1.00
0.1 0 0 0.00 0.00 1.00 1.00
v0.2 0 0 0.00 0.00 1.00 1.00
0.3 0 0 0.00 0.00 1.00 1.00
0.4 3 0 0.25 0.00 1.00 1.00
0.5 4 0 0.33 0.00 0.75 1.00
0.6 2 1 0.17 0.08 0.42 1.00
0.7 3 2 0.25 0.17 0.25 0.92
0.8 0 3 0.00 0.25 0.00 0.75
0.9 0 3 0.00 0.25 0.00 0.50
1.0 0 3 0.00 0.25 0.00 0.25
TOTAL 12 12 1.00 1.00 5.42 8.42
EXPERTIZED VALUES 0.54 0.84
Source: own elaboration
In a similar way, this procedure is carried out for all the objectives set, in order to determine the most suitable objectives, which are considered as approved. For this purpose, the thresholds determined in the first expert appraisal must be within the confidence band or interval to be accepted; otherwise, they will not be taken into consideration; the approved ones will be the objectives pursued by the organization through the ZBB. Considering the idea of Agner (2020) in the budget area, it is key to designate a dedicated budget analyst to manage and monitor the transition and activation budget cost center(s) in order to coordinate resources and schedule staff. It will also produce consistent budget tracking and reporting (Table VIII).
Table VIII. Objectives Ap Droval
N° Objectives Threshold INTERVALS RESULTS
1 Train personnel in the technological field. 0.827 [ 0.54 ; 0.84 ] Approved
2 Increase production margin. 0.833 [ 0.67 ; 0.89 ] Approved
3 Increase market share. 0.593 [ 0.62 ; 0.88 ] Denied
4 Orient marketing processes to new market niches. 0.813 [ 0.54 ; 0.85 ] Approved
5 Incorporate biosafety elements in the work environment. 0.760 [ 0.71 ; 0.85 ] Approved
6 Launch products with new designs 0.647 [ 0.73 ; 0.89 ] Denied
7 Optimize teleworking 0.880 [ 0.82 ; 0.90 ] Approved
8 Apply for credit financing from public or private banks. 0.793 [ 0.74 ; 0.96 ] Approved
9 Create innovative management models 0.593 [ 0.65 0.79 ] Denied
10 Modernize after-sales service 0.820 [ 0.80 0.93 ] Approved
11 Use social networks for better product positioning 0.833 [ 0.81 0.95 ] Approved
12 Improve return on investment 0.573 [ 0.64 0.89 ] Denied
13 Increase customer satisfaction 0.613 [ 0.73 0.86 ] Denied
14 Optimize costs and expenses 0.813 [ 0.74 0.91 ] Approved
15 Improve company image 0.540 [ 0.66 0.81 ] Denied
16 Promote job stability 0.613 [ 0.72 0.93 ] Denied
Source: own elaboration
When projecting the ZBB with TpFN, it is structured based on the approved objectives (Table VII). These were determined considering the most important priorities of the company in order to achieve economic reactivation. Subsequently, the economic resources to be delivered for the fulfillment of these objectives are defined. The budget mentioned above is reported in Table IX.
Table IX. Budget allocation by objective
№ Delineated Objectives Budget allocation (US$)
1 Train personnel in the technological field. 3,200
2 Increase the production margin. 7,300
3 Orient marketing processes to new market niches. 2,500
4 Incorporate biosafety elements in the work environment. 3,800
5 Optimize teleworking. 1,300
6 Apply for credit financing from public or private banks. 1,000
7 Modernize after-sales service 3,200
8 Use social networks to better position the product. 2,000
9 Optimize costs and expenses 1,200
TOTAL 25,500
Source: own elaboration
The total budgeted value is determined as the sum of the budget allocation of all the objectives to be met. This value amounts to US$25,500. The financial department is contacted to determine the revenues that the company would generate at the beginning of the period. Management, based on the estimated revenues, establishes a pessimistic position of US$17,159 and an optimistic one of US$24,000 for meeting the objectives set (Table X).
Table X. Estimated revenues
REVENUES PESSIMISTIC OPTIMISTIC POSITION (US$)
COMPONENTS POSITION (US$)
Unit sales 12,530 14,500
Short-term collections 2,340 3,700
Long-term collections 1,521 2,550
Other income 768 3,250
TOTALS 17,159 24,000
Source: own elaboration
To represent the organization's budget levels, the TrFN are established, expressed by (a1, [a2, a3], a4), where a1 = lower end; [a2, a3] = maximum assumption; a4 = upper end (Figure I).
Figure I. Trapezoidal fuzzy number
Source: own elaboration
From Table IX, related to the income that the company would generate, the pessimistic position refers to the certainty of being able to invest in each of the defined objectives. On the other hand, the optimistic position refers to the efforts that will be possible to achieve these objectives. The former takes a valuation related to unity (1) as opposed to the other whose allocation will be zero (0), if it exceeds this value (Figure II).
Figure II. Estimated economic resources
Source: own elaboration
In order to determine the budget levels with their respective assigned items (lower end, maximum assumption, upper end), it is necessary that the experts in the financial area assign these economic values to achieve the objectives set (Table XI).
Table XI. Budget levels
LEVELS
LEVEL 1 LEVEL 2 LEVEL 3 LEVEL 4 LEVEL 5 LEVEL 6 LEVEL 7 LEVEL 8 LEVEL 9
ADDING OF LEVELS
Ai
Ai +A2
A± + Az + A3 + A+ + A5 + A^ + A7
Aj. + A2 + A3 + A4 + As + A^ + A7 + As
A± + A2 + A3 + A+ + A5 + A6 + A7 + As + A5
TrFN
ai a2 a3 a4
(US$)
3,450 3,536 3,590 3,690
5,278 5,893 6,043 6,130
7,032 7,345 7,560 7,890
8,340 8,620 8,900 9,120
10,230 10,530 11,450 12,400
14,230 14,900 15,200 16,230
17,800 18,300 19,100 19,500
20,450 21,300 21,900 23,000
25,000 26,400 28,400 30,120
Source: own elaboration
A geometric summary of the nine budgeted levels, which are based on the objectives established for the companies, is presented. They range from Level 1 with an investment of (3,450; [3,536 ; 3,590]; 3,690), US$, to level 9 whose funding will be (25,000; [26,400 ; 28,400]; 30,120), US$. By means of a geometric trace of TrFN, trapezoids are found that identify the budget levels for each objective to be achieved and the corresponding budget constraint. Levels 1 to 6 are accepted, since the budget covers the economic items, while the company's managers, according to their level of coverage, will analyze Levels 7 and 8 while Level 9 will be rejected for not having budget coverage (Figure III).
Figure III. Budgetary level analysis
Source: own elaboration
To determine the lack of coverage index in relation to Level 7, we proceed to calculate geometrically the areas of the trapezoids with the help of GeoGebra software. To do this, the intersection points E and F are found, then the area of the trapezoid ABEF and the trapezoid ABCD is determined, and the coefficient between them, which is called IFC, is determined. Subsequently, the coverage index expressed by IC = 1 - IFC is calculated. In the case of this level, the result is 84.45%, which is a high percentage, so management should approve it.
Lack of coverage index ( IFC ) Trapezoid area ABEF Trapezoid area ABCD
'FC
IFC
186.76
= 0.1555
1200.5 Coverage index ( Ic )
FC
Ic = 84.45%
The same procedure is used to determine the coverage index for Level 8. The intersection points K and L are found, the area of the trapezoid GHKL and of the trapezoid GHIJ are calculated, and similarly the CFI and CI are determined, the result of which is 46.62%. In this case, it is left to management to make the decision to approve or reject this budget level.
Lack of coverage index ( IFC )
Trapezoid area GHKL Trapezoid area GHIJ
IFC = 53.33% Coverage index ( Ic )
Ir = 46.62%
5. Discussion
The objective of the research is proposed a methodology to prepare a ZBB for SMEs in Ecuador, applying fuzzy logic. As can be seen, with the support of the fuzzy logic technique of expertise, the most suitable objectives that could be achieved by the industrial company of Cuenca (Ecuador) were delimited. Based on this, the ZBB is structured, determining budget levels to achieve as a whole the fulfillment of objectives.
The budget constraint that the organization will have for the following accounting periods was determined with the support of management. This restriction was placed in a pessimistic position of US$17,159 and an optimistic position of US$24,000, in order to meet the defined objectives, it being evident that any value higher than the optimistic position is difficult to meet.
Budget levels 1 to 9 have been structured, with appropriately allocated economic items. Management accepts levels 1 to 6 because they are within the level of coverage. They will cover the necessary requirements to achieve the following objectives: A1: Train personnel in technology; A2: Increase the production margin; A3: Orient marketing processes to new market niches; A4: Incorporate biosecurity elements in the work environment; A5: Optimize teleworking; and, A6: Request financing through loans from public or private banks.
Budget Level 7, with a coverage rate of 84.45%, provides a good margin of compliance to achieve the defined objectives and should therefore be approved by management. The coverage rate for Level 8 is 46.62%, which is a very limited percentage of coverage, and the decision to approve or reject it will be at management's discretion. Level 9 will be rejected because it does not have budgetary coverage.
The application of the TpFN by means of a geometric trace, allows the identification of the budget levels to be reached and the corresponding budget restriction; with this, the company will be able to comply with the delimited objectives to optimize the decision-making processes. This technique represents a valuable tool for SMEs, whose purpose is to analyze, evaluate and allocate economic items in a real and efficient way to the objectives set. It will be possible to visualize an impact on the managers, who will be able to improve the different areas of their company, by means of a more efficient decision-making, favoring its economic reactivation in times of health crisis.
6. Conclusions
SMEs in the industrial sector of Cuenca (Ecuador) must develop work methodologies to activate their operations in recessionary periods and contingency situations. Adopt and adapt new methodologies and work methods, giving priority to what is essentially necessary to keep their operations active and not perish in the attempt; in many situations, starting from scratch, without any preconceptions, was necessary. Structuring a new methodology for preparing a budget makes it possible to meet the expectations set for the following period, i.e., with new processes different from the usual ones within the company.
The ZBB provides a means of meeting the defined objectives, with the purpose of improving decision making and proposing the necessary measures to solve the problems, taking advantage of the resources in an efficient way.
The ZBB, supported with fuzzy logic, incorporates the opinion of experts in the financial area and managers of SMEs, in order to make efficient use in the planning of financial resources, where a thorough analysis is warranted for the assessment and approval of budget levels within a systematization of aspects that encompass objectives whose purpose is to direct the organization to the achievement of its ideals.
The research demonstrated the importance of applying the technique of expertise, typical of fuzzy logic, since it is possible to select the ideal and feasible objectives to achieve the goals set. This makes it possible to reduce uncertainty, trapping the organizational economic dynamics aimed at achieving the objectives, to which economic resources are assigned for execution in the budgeted period. Further research can extend the subject and address related business sectors in other Latin American realities. That comparative studies can be projected and validate the proposed methodology, combining ZBB with support for fuzzy logic in SMEs.
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