**3. Hybrid model for optimization**

**1. Introduction**

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agement process.

programs and the current legislation.

conclusions and future works.

**2. The budget guidelines law**

Participatory management establishes a dynamic of democratization of the public administration, since it associates planning and popular participation, through political definitions and adjustments and adjustments in the process, when necessary. Its purpose is to discuss and define the population, in a democratic way, the distribution of investment resources. Exactly because it is the result of the discussion and the decisions of the society, the works or investments prioritized, in general, involve the most complex problems of the communities. They are works that require more detailed projects and careful executions. Significant challenges put pressure on governments at the federal, state, and municipal levels, requiring them to be more creative and effective ways of achieving results, from the employment of increasingly

Considering that the process of managing public resources is defined in the budget, it is necessary that it contains all the elements that facilitate and allow the correct application of these, among the universe of interests presented by the population through the participatory man-

Within this context, we present a methodology for modeling the problem that seeks in science based on multicriteria and linear programming the optimal solution for its proposition for the Public Budget, whose objective is to provide the public administrator with the necessary tools to optimize the application of the available public resources, considering the environment in which it is inserted. To determine the optimal way of applying the funds authorized by the Legislative Power, in order to maintain the administrative machinery, enforce the laws and still meet the desires of the population with increasingly scarce financial resources. Our proposal is to apply these technologies to public finances, whose objective is to model the public budget in order to maximize the available resources, taking into account the government

In the Section 2, presents Budget Guidelines Law. Moreover, Section 3, is given Hybrid Model for Optimization. More details of the Multicriteria Modeling provided is Section 4. On the other hand, in Section 5 provides more information on the applied methodologies. Furthermore, in Section 6 the computational results obtained using. Finally, Section 6 presents

In the Budget Guidelines Law (BGL), expenditure is classified through several criteria, the most important of which are functions, programs, nature of expenditure and elements, which leads to a more complex and more difficult codification, different interpretations for some of its components. The Law requires that some principles be observed in the execution of the budget, especially flexibility, that is, adaptation to new or unforeseeable situations. This flexibility necessary for public managers is extremely limited due to the restrictions imposed by

scarce resources and an increasingly demanding and participatory population.

The proposed model is composed of two steps: initially to construct the objective function from the multicriteria analysis [1–3]. Next, the construction and solution of a linear programming model based on the objective function constructed in the first phase and the legal, minimum and maximum constraints defined for each variable to be calculated, called budget allocations.

#### **3.1. Model in linear programming**

The aim is to maximize the actions to be carried out through budget allocations, considering the degree of importance of these actions (Government Programs). Budget allocations are defined as variables x<sup>i</sup> .

The best distribution of the budget will be the one that better attends to the actions of the Government Programs, considering the importance defined by the widespread participation coordinated by the administration in question. The optimization criterion is then the sum of the budget allocations of each activity weighted by the degree of importance of the Government Program to which it is linked. We can stand for the objective function in Eq. (1),

$$\max \sum\_{i=1}^{n} k\_i \mathbf{x}\_i \tag{1}$$

x<sup>i</sup> = Budget allocation of the activity related to the Government Program i;

ki = Degree of importance of the Government Program i;

n = Number of government program activities in the budget.

The evaluation of the pertinent legislation Budget Guidelines Law (BGL), and Fiscal Responsibility Law (FRL) allowed to find some legal restrictions, which we present below. The definitions given are merely illustrative for the assembly of the model; In their application, should follow what is recommended in the specific literature.

#### **3.2. Restrictions**

I. Expenditures may only be made up to the limit of the budget credits and additional credits, and following the disbursement schedule of the Unit, duly approved in Eq. (2),

$$\text{Restriction 1: } \sum\_{i=1}^{n} x\_i = \text{RCTot} \tag{2}$$

n = Number of government program activities in the budget;

RCTot = Total revenue previously defined by the finance team, according to the pertinent legislation;

II. According to Eq. (3) for the FRL, personnel expenses are the sum of the expenses of the Federation entity with the assets; The expenses for inactive and pensioners; Fixed and variable salaries and benefits; The subsidies, retirement benefits; Reforms and pensions; The added ones of any nature; The gratuities, overtime and personal advantages; Social charges; The contributions collected by to private pension entities. The FRL limits spending on public sector personnel to 60% of net revenue for states and municipalities. In the municipal sphere, the 60% limit will be distributed among the Legislative Branch: 6% (including the Municipal Court of Audit) and Executive Branch: 54%.

$$\text{Restriction 2: } \sum\_{i=p\_i}^{p} \mathbf{x}\_i \le 0.495^\circ \text{ RCL} \tag{3}$$

p1 …pn = activities of Government Programs in the budget related to personnel;

RCL = Net revenue.

III and IV. According to Eq. (4) and Eq. (5) for the Federal Constitution, the municipality must distribute to education, not less than 25% of its tax collection and transfers. Of these 25%, 60% should be earmarked for primary school funding and the remaining 40% for other levels of education. In addition, 60% of the resources should be earmarked exclusively for the payment of teachers' salaries, with a view to enhancing the teaching profession and improving the quality of teaching.

$$\text{Restriction 3:} \sum\_{\mathbf{x}\_i}^{\epsilon\_\*} \mathbf{x}\_i \ge 0.25 \times \text{RCE} \tag{4}$$

$$\text{Restriction 4: } \sum\_{i \text{ find } i}^{\text{fuel}\_i} x\_i \ge 0.25 \times 0.60 \times \text{RCFund} \tag{5}$$

e1 , …, en = activities of Government Programs in the education budget;

The evaluation of the pertinent legislation Budget Guidelines Law (BGL), and Fiscal Responsibility Law (FRL) allowed to find some legal restrictions, which we present below. The definitions given are merely illustrative for the assembly of the model; In their application, should

I. Expenditures may only be made up to the limit of the budget credits and additional credits,

*i*=1 *n*

RCTot = Total revenue previously defined by the finance team, according to the pertinent

II. According to Eq. (3) for the FRL, personnel expenses are the sum of the expenses of the Federation entity with the assets; The expenses for inactive and pensioners; Fixed and variable salaries and benefits; The subsidies, retirement benefits; Reforms and pensions; The added ones of any nature; The gratuities, overtime and personal advantages; Social charges; The contributions collected by to private pension entities. The FRL limits spending on public sector personnel to 60% of net revenue for states and municipalities. In the municipal sphere, the 60% limit will be distributed among the Legislative Branch: 6% (including the Municipal

> *i*=*p*<sup>1</sup> *pn*

III and IV. According to Eq. (4) and Eq. (5) for the Federal Constitution, the municipality must distribute to education, not less than 25% of its tax collection and transfers. Of these 25%, 60% should be earmarked for primary school funding and the remaining 40% for other levels of education. In addition, 60% of the resources should be earmarked exclusively for the payment of teachers' salaries, with a view to enhancing the teaching profession and improving

> *i*=*e*<sup>1</sup> *en*

*i*=*fund*<sup>1</sup> *fundn*

…pn = activities of Government Programs in the budget related to personnel;

*xi* = RCTot (2)

x*<sup>i</sup>* ≤ 0.495<sup>∗</sup> RCL (3)

x*<sup>i</sup>* ≥ 0.25 × RCE (4)

*xi* ≥ 0.25 × 0.60 × RCFund (5)

and following the disbursement schedule of the Unit, duly approved in Eq. (2),

follow what is recommended in the specific literature.

Restriction **1**: ∑

Court of Audit) and Executive Branch: 54%.

Restriction 2: ∑

Restriction 3: ∑

Restriction 4: ∑

n = Number of government program activities in the budget;

**3.2. Restrictions**

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legislation;

p1

RCL = Net revenue.

the quality of teaching.

fund1 ,…fundn = activities of Government Programs included in the budget for primary education;

RCE = income for education, defined in the Budget Guidelines Law (BGL);

RCFund = revenue for calculation of fundamental education, defined in the BGL.

V. Expenditure on health: the linking percentage is 15% of its tax revenue and constitutional transfers in Eq. (6). This rate must be reached within 5 years from the entry into force of the Amendment. In their first year, percentages below 15% for health financing should be increased gradually by the 5th year after their approval, reducing that gap by at least 1/5 per year. For modeling purposes, we are considering only the percentage of binding of 15%, understanding that the Administration in question is with this percentage stabilized.

$$\text{Restriction 5: } \sum\_{l=\nu\_i}^{\epsilon} \mathbf{x}\_l \ge 0.15^\circ \text{ RCS} \tag{6}$$

s1 …sn = government Program activities in the health budget;

RCS = For calculation of Health, defined in the BGL.

VI. The budget distributed to the City Council should be at most equal to 5% of tax revenue in Eq. (7).

$$\text{Restriction 6: } \sum\_{i=\epsilon\_i}^{\epsilon} \mathbf{x}\_i \le 0.05^\circ \text{ RCT} \tag{7}$$

c1 … cn = number of activities of Government Programs gave in the budget relative to City Hall;

RCT = Tax revenue, as defined in the BGL.

VII. In Eq. (8) to current expenditure cannot be financed by capital expenditure; That is, the value of current expenses must be less than or equal to the value of current revenues. Current spending is the cost of supporting public services; Without any return to equity of equal value. Capital expenditures are expenses where there is the reward for the direct entrance of a kind, of corresponding value, in the patrimony; that is, investments, financial investments, and capital transfers.

$$\text{Restriction } \mathsf{T} \colon \sum\_{l=d\_1}^{d} \mathsf{x}\_l \le \mathsf{RC} \tag{8}$$

dc1 …dcn = current expenses supported in the BGL;

RC = Current revenues (revenues directly from the Public Administration that are directly or indirectly consumed in expenses with the maintenance of public services previously created - do not include capital funds).

VIII. Finally, in Eq. (9) all Government Programs to which the budget allocations (model variables) are linked have a certain political importance for Administration (index k of the model); and, so, they must be positive, different from zero, greater of a value u and smaller of a value d, both decided by the Administration; or in the range:

$$\mathbf{u}\_{l} \le \mathbf{x}\_{l} \le \mathbf{d}\_{l} \forall \mathbf{x}\_{l} \tag{9}$$
