**4.1. Range of variation of fi**

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

The degrees of importance of government programs should be defined based on the multicriteria methodology, defined as k in the model, will define the maximum value of the objective function and will be a decisive factor in the definition of controlled variables of the model. The

• The revenues, included in our model, that restrict the value of the controlled variables;

• The relation of the activities of the Government Programs that will define the size of the vector of controlled variables. Government Programs will be treated as constants in the

• The classification of the activities of the Government Programs, through Participatory Management, duly reviewed by the Principal Manager. This classification will be the input

In addition, each government program is divided into activities and each of these activities should be classified into two levels: initially classifies the activity by fundamental values such as Health, Education, Environment, etc. Moreover, specifies each fundamental value, called here sub-criteria, creating a two-level classification. The value tree will be built with as many fundamental values as believed necessary by management, including the participation of society (fundamental value A, B, C, D, etc. …). Each fundamental value will have as many sub criteria as are defined by management and society (sub criteria I, II, III, etc. …). The proposed modeling considers that the software Measuring Attractiveness by a Categorical Based Evaluation Technique (M-MACBETH): method elaborated by [5, 6]) is applied to define the degrees of importance, from the tree of Values built. The attractiveness indexes of these fundamental values among themselves and the sub-criteria should be giving by consensus

Furthermore, a matrix of fundamental values (A, B, C, …) is created and the attractiveness of them is compared to each other, generating the index f of each of these values by the M-MACBETH method. The index f is the result of comparability of the first level of the value tree. Each fundamental value will have sub-criteria linked to it, which should also be compared to each other. A new value matrix is constructed for each fundamental value, comparing the attractiveness of its sub criteria, generating the index s of each sub-criterion by the

to calculate the degree of importance of each Government Program.

within a participatory management program of the actors.

M-MACBETH method (second level index of comparability).

∀x*<sup>i</sup>* (9)

d, both decided by the Administration; or in the range:

information required for the model to be provided [4]:

u*<sup>i</sup>* ≤ x*<sup>i</sup>* ≤ d*<sup>i</sup>*

**4. Multicriteria modeling**

188 Management of Information Systems

model;

The actors decide which fundamental value has the greatest and the least importance, they then receive grades 100 and 10, respectively. These levels of attractiveness, because they are defined as major and minor, by the actors must receive the maximum and minimum values, and therefore cannot assume intermediate values; Consequently, there is no variation interval.

#### **4.2. Range of variation of s**

For each fundamental value the actors' levels of attractiveness were defined by their sub criteria. As in the valuation of f, in each case there is an item of greater and another of lesser importance. These items receive degrees 100 and 10, respectively. These attractiveness levels, since they are defined as major and minor by the actors, must receive the maximum and minimum values, and therefore cannot assume intermediate values; Consequently, there is no range of variation.

Furthermore, each parcel f and s that composes index k can assume a range of values, without, however, changing the level of attractiveness that is calculated by M-MACBETH. We call this range of "Interval" values. The M-MACBETH presents the ranges of variation that the f and s indexes generated by it can assume, guaranteeing the valuation of the levels of attractiveness defined by the actors.

A range of variation for the index k, defined from the solution of the problem in linear programming, considered objective function and constraints. Let's call this interval "Interval θ ".

Moreover, each step of our proposal generates a set of values in which the degrees of importance of government activities (k) may be inserted: the intervals β and θ. The intersection of these sets stands for the interval at which, in fact, our objective function indexes k may vary, the "Interval γ".

#### **4.3. Variable time**

The construction phase of the model supports the analysis of the time horizons of the proposed modeling. The budget part is a public instrument defined for the fiscal year (fiscal year - 12 months). Your legal restrictions must be respected at the end of each exercise. This hybrid model proposes a method for the construction of the budget piece for a given fiscal year, which guarantees compliance with the legislation, and other restrictions imposed. It can also be used in budget supplementation, during the exercise, to verify compliance with legislation; Since, once the restrictions have been respected throughout the year, we will be guaranteeing compliance with these restrictions at the end of the exercise. But in this case, we must evaluate the results in time every round of the model.

### **4.4. Duality analysis**

The primal of the proposed model has as goal function the sum of budget allocations that are weighted by the degree of importance of the Government Program to each of them associated, degrees these, defined from multicriteria analysis by popular participation. Its restrictions are the legal ones regarding the revenue and the minimum and maximum limits defined by the public administration for each activity. These values of revenue and maximum and minimum limits are constant values in our proposal, defined at the time of the elaboration of the primal model. They change the distribution of the budget: if, for example, Total Revenue is changed, the values of the budget allocations will change. If the minimum or maximum amount to be applied in the school lunch activity is changed, this will have a greater or lesser impact than a variation in Total Revenue.

Consider:

z = number of restrictions on budget allocations;

w = number of legal restrictions;

r<sup>j</sup> = values defined as restrictive revenue from the applications of budget allocations;

p<sup>j</sup> = values determined as lower/upper limits of the budget allocations for the activities of the Government Programs;

y<sup>j</sup> = degree of effect about values defined by the public administration as lower/upper limits and revenues in the budget distribution.

On the other hand, restrictive revenues (rj ) are those present in the constraints of the primal as legal impositions. Some constraints of the primal are formed by equations that impose to the budget allocations x<sup>i</sup> minimum limit, others impose maximum limit. These minimum or maximum values determined by administration at the time of constructing the primal are pj .

By "degree of impact" we mean the relative importance that these lower (upper) limits on the main problem have in defining budget allocations. That is, yj measures the relative impact of that arbitrariness of the lower value (or higher) and of the revenue in the budget distribution. We can present in Eq. (10) the duality of the proposed model in the form:

$$\text{Dual objective function:} \text{Min } \sum\_{j=1}^{w} r\_j y\_j + \sum\_{j=w+1}^{z} p\_j y\_j \tag{10}$$

The values "pj " and "rj " are "arbitrated" in the primal, that is, they are constant values defined at the time of the elaboration of the primal model. Once altered they affect, in some way, the solution of the primal, that is, the distribution of the budget.

The purpose of the dual is to MINIMIZE the impacts of the "arbitrated" upper (upper) limits and legal restrictions on revenues in budget distribution.

The greater the degree of effect y, the greater the impact of the arbitrated limits, or the revenue, on the distribution. That is, a change to a greater or lesser extent in the coefficient going with the y in the objective function of the dual will have a greater or lesser impact, respectively, in the distribution of budget allocations.

Each constraint of the dual will be formed by an inequality where a k will be present in its second term, which will stand for them by in Eq. (11):

$$\sum\_{j=1}^{n} q\_{\iota\_{\iota\_{\ast}}} y\_j \ge k\_{\iota'} \mathbf{1} \le \mathbf{n} \le \mathbf{v} \tag{11}$$

at where:

**4.4. Duality analysis**

190 Management of Information Systems

a variation in Total Revenue.

w = number of legal restrictions;

and revenues in the budget distribution.

On the other hand, restrictive revenues (rj

" and "rj

main problem have in defining budget allocations. That is, yj

Dual objective function:Min ∑

solution of the primal, that is, the distribution of the budget.

and legal restrictions on revenues in budget distribution.

We can present in Eq. (10) the duality of the proposed model in the form:

Government Programs;

the budget allocations x<sup>i</sup>

The values "pj

z = number of restrictions on budget allocations;

Consider:

The primal of the proposed model has as goal function the sum of budget allocations that are weighted by the degree of importance of the Government Program to each of them associated, degrees these, defined from multicriteria analysis by popular participation. Its restrictions are the legal ones regarding the revenue and the minimum and maximum limits defined by the public administration for each activity. These values of revenue and maximum and minimum limits are constant values in our proposal, defined at the time of the elaboration of the primal model. They change the distribution of the budget: if, for example, Total Revenue is changed, the values of the budget allocations will change. If the minimum or maximum amount to be applied in the school lunch activity is changed, this will have a greater or lesser impact than

r<sup>j</sup> = values defined as restrictive revenue from the applications of budget allocations;

p<sup>j</sup> = values determined as lower/upper limits of the budget allocations for the activities of the

y<sup>j</sup> = degree of effect about values defined by the public administration as lower/upper limits

as legal impositions. Some constraints of the primal are formed by equations that impose to

maximum values determined by administration at the time of constructing the primal are pj

By "degree of impact" we mean the relative importance that these lower (upper) limits on the

that arbitrariness of the lower value (or higher) and of the revenue in the budget distribution.

at the time of the elaboration of the primal model. Once altered they affect, in some way, the

The purpose of the dual is to MINIMIZE the impacts of the "arbitrated" upper (upper) limits

The greater the degree of effect y, the greater the impact of the arbitrated limits, or the revenue, on the distribution. That is, a change to a greater or lesser extent in the coefficient going with

*j*=1 *w*

) are those present in the constraints of the primal

measures the relative impact of

*pj yj* (10)

.

minimum limit, others impose maximum limit. These minimum or

*rj yj* + ∑ *j*=*w*+1 *z*

" are "arbitrated" in the primal, that is, they are constant values defined

m = number of constraints of the budget distribution;

v = number of variables budget allocations;

q = is the coefficient of x in the nth budget constraint.
