In general, decisions are made either individually or collectively. For Kocher and Sutter (2005), in individual choices, to achieve the aimed objectives, one starts from the confrontation of the preferences, alternatives, and restrictions, in search of individual benefit. However, in group decisions, one seeks a consensual choice based on individual preferences ( Hammond et al., 1999).

According to Bregalda (2017), more sophisticated techniques and models have been built, such as Pareto-optimal, developed to address the multiobjective problems, with the most feasible solutions possible. Pareto’s efficient solution can be obtained so that the chosen alternative achieves a broad value in all criteria and does not have a simultaneous decrease – a value dominated by another alternative ( Gomes & Gomes, 2019; Silva 2020).

The Maut is a discrete method for having a discrete number of discrete alternatives. It is used to determine the importance attributed to a certain criterion over another and to prioritize alternatives. In general, multiattribute methods refer to methods for selecting, ordering, or categorizing among a finite number of alternatives, explicitly known ( Clímaco et al., 1996). As part of these optimization problems, besides the Maut method, the Analytical Hierarchy Process (AHP) and the Quality Function Deployment (QFD) have been used to apply multiple decision criteria ( Matsuada et al., 2000).

In a multiple criteria context, according to Zopounidis and Doumpos (2002), decision-making problems are carried out in the following paradigm: a set A of alternatives (e.g., companies, investment projects, and portfolios) is considered, and an attempt is made to make an optimal decision considering all relevant factors to the analysis. Since these factors usually lead to conflicting results and conclusions, the optimal decision is not ideal from the traditional optimization perspective.

In Maut, the objective function is the mathematical representation of the efficiency criteria applied to the optimization problem. It is influenced by the project variables, known as decision variables of the problem ( Gomes & Gomes, 2019). The solution space consists of all points that satisfy the problem’s constraints. The optimal solution in the maximization problem corresponds to the point of maximum value for the objective function ( Gomes & Gomes, 2019).

For Silva et al. (2015), the EE approach observes the optimal moment of equipment replacement as the starting point of the concepts of useful life and economic life of a good. The equivalent uniform annual cost method proposes that the economic life of an equipment corresponds to the period in which this cost is minimal and, therefore, the optimal moment for its replacement ( Silva et al., 2015).

In the evaluation of an equipment’s economic life, the analysts should use techniques that consider the money’s value over a time scale to recognize opportunities for positive outcomes when estimating the series of expected cash flows associated with the alternatives ( Lima et al., 2015).

The importance of costs whether for use or ownership is observed in Souza et al. (2015), defining TCO as a complex approach in which the buying organization needs to identify all costs considered relevant in the activities of acquisition, possession, and use of a good or service, quantified for each supplier. According to Onkham et al. (2012), TCO provides metrics to assess the costs of the entire life cycle of a product by considering, besides the acquisition value, the costs associated with its use and disposal.

Thus, TCO is a tool to support strategic cost management in purchasing decisions ( Ellram & Siferd, 1998). In this way, Diniz and Paixão (2017) compared the costs of own and outsourced vehicle fleets in commercial operations of a private company, verifying, through a projected scenario, that fleet ownership was more profitable. The application of the TCO analysis contributed to demonstrating the hidden costs, which are not considered in the economic evaluation of the equipment, which could have changed the purchase decision in a traditional analysis applied without the tool ( Coser & Souza, 2017).

Using a vehicle as a product, “TCO covers all expenses accrued by a vehicle owner, including a one-time purchase cost and other costs such as fuel, taxes, maintenance, and repair” ( Redelbach et al., 2012, p. 2).

Using the Maut to build optimal solutions, management indicators are defined. Each fleet management indicator is an attribute, such as having agility in fleet adaptation, speed in the replacement of cars with accidents, among other relevant factors for good management efficiency. Decision variables, related to a better fleet management capacity, are defined as relevant management aspects for the decision-maker. Each decision variable will be composed of a set of attributes of management indicators. The two decision variables are:

1. Xª = Fleet maintenance

   Yª = Fleet availability

Each objective function with its respective decision variables has its attributes that belong to the decision vector ( Table 1). These attributes were selected from the literature on positive factors of vehicle leasing and ratified as relevant by the managers of the Public Safety Department of Ceará state government. The subjective preferences of decision-makers, among the alternatives, are measured or revealed by the weighting of some criteria. Each rental company that disputes the preference of the decision-maker will have its Xª and Yª measured. The objective functions represent the decisionmaker’s preferences among the attributes of a set. The decision variables, Xª and Yª, are equivalent to the sum of the weights of attributes x _i and y _i, as shown in Table 1.

The decision variables refer to the decisions to be made aiming to find the solution to the problem. To parameterize these variables, attributes will be defined according to management needs, based on the following assumptions:

1. 1. Fleet outsourcing companies observe the definitions of the management parameters set out in the bidding and set their prices according to their capacities and costs.

   It is assumed that companies compete in a perfect market and that there is no corruption in bidding to misrepresent the market price.

   Given prices in a competitive market, the proposals from outsourcing companies will have their prices directly proportional to the degree of quality imposed for each decision variable (management), connected to the attributes defined in the bid notice.

Considering the assumptions, the manager parameters the decision variables with the attributes listed in Table 1 and weights criteria according to Table 2. This is an a priori decision model, the manager is consulted only once, before the optimization process begins, and the information obtained regarding their interests is used to guide the search for the favorite solution belonging to the Pareto frontier.

The decision-makers defined the attributes in Table 1 with the weights for each criterion according to Table 2, creating, from the sum of the attributes for each decision variable, Xª and Yª.

Once each attribute of the decision variables has been defined, the process now is to define the scoring of the alternatives, therefore, the companies (alternatives) that are competing will be evaluated and scored for each attribute, using the weights in Table 2. The final score of the company is given by the final sum of the attributes of each decision variable, with the respective score ranging from 0 to 10 in Xª and Yª.

Given the definitions of the weights ( Table 2) on the attributes in Table 1, Table 3 is used to reclassify the weights into scores from scaling intervals of the sum of the attributes by objective function, f(Xª) and f(Yª). This is a discrete decision model since it has a finite number of alternatives.

There are two decision variables, Xª and Yª, with three scores (0, 1, and 2), as presented in Table 3, from the weights on the attributes in Table 1. The order (Xª, Yª) is relevant. Considering Xª = 2 and Yª = 1, then there is the point (2, 1), which is different from (1, 2). Since the independent attributes support the two variables (Xª and Yª) with difference in order are repeatable (1, 1), the problem will have nine alternatives, given the arrangement with repetition A _{(n, p)} = n ^p.

The multicriteria decision model is presented by the set of equations from (1) to (17). Z represents the additive utility function revealing the decision-makers’ preferences by means of the management quality parameters, used to decide which company will be accepted given the change from own fleet to leased fleet (outsourced). Functions (1) and (2) are the objective functions, constant functions according to intervals for the decision variables Xª and Yª of the alternative A. Therefore, the optimization problem is configured as follows:

• Maximize:

  $Z_x^a=f\left(X^a\right)=\{\begin{array}{c}0if0<X^a\le 4\\ 1if4<X^a\le 7\\ 2if7<X^a\le 10\end{array}$(1)
  $Z_x^a=f\left(X^a\right)=\{\begin{array}{c}0if0<Y^a\le 4\\ 1if4<Y^a\le 7\\ 2if7<Y^a\le 10\end{array}$(2)
  $Z^a=\left(Z_x^a,Z_y^a\right)$(3)
  $Z^a=\left(f\left(X^a\right),f\left(Y^a\right)\right)$(4)
  $Z^a=f\left(X^a\right)+f\left(Y^a\right)=Z_x^a+Z_y^a$(5)

  Subject to restrictions:

  $0<X^a,Y^a\le 10$(6)
  $X^a+Y^a\ge 13\forall X^a,Y^a\ge 5$(7)
  $Z^a=f\left(X^a\right)+f\left(Y^a\right)=Z_x^a+Z_z^a\ge 3\forall Z_x^a,Z_y^a>0$(8)
  $d_i=\left(X_x^a,Y_y^a\right)\in D=\text{Decisionspace}$(9)
  $Z^a=\left(Z_x^a,Z_y^a\right)\in Z=\text{Objectivespace}$(10)
  $Z=Z_x^a+Z_y^a\ge 3$(11)
  $Z_x^a,Z_y^a\in N/0\le Z_x^a,Z_y^a\le 2$(12)
  $p_x{Z}_x^a+p_y{Z}_y^a\le L_e$(13)
  $\Delta z_i\to \Delta Costsof{z}_i=\Delta p_i{z}_i$(14)
  $x_i,y_i,\in N/0\le x_i,y_i\le 2$(15)

Considering:

Each vector d _i (9) in the decision space domain D will have as its image a vector z from the objective space Z. The search space D is the domain (bounded or unbounded) that contains the parameter values. It corresponds to the solution space. The dimension of the search space is defined by the number of parameters involved in the solutions – for example, if each solution is formed by three parameters, the search space is three-dimensional.

For each solution $d_i=\left(X_x^a,Y_y^a\right)$ in D, there is a point $Z^a=\left(Z_x^a,Z_y^a\right)$ in the objective space Z, as shown in Table 3. In the case of conflicting objectives, when the optimization of one of the objectives causes the deterioration of the other objectives, single-objective optimization is not sufficient. In multiobjective optimization, the concept of optimality is based on the concept of Pareto dominance. In this particular problem, the two objective functions specify fleet management criteria where increasing the quality and quantity of the respective functions raises the company’s costs. Since they will be in a competitive process, it is expected that they will take their offers in terms of quality and quantity of attributes to the boundary, therefore, to increase X they will have to sacrifice Y and vice-versa.

The concept of optimal solution is replaced by the concept of efficiency, which is related to the concept of non-dominance, the former being associated with the solutions space (decision) and the latter with the objectives space (criteria). The decision-maker’s preference information is obtained through the choice of the objective function leading to an optimization.

The restriction (7) in the decision space is equivalent to restriction (11) in the objectives space, meaning that the manager defined that the possibilities of acceptance for changing from own to leased fleet will occur if the variables of the attributes, summarized by Table 3, have the sum of the weights of, at least, the criteria three, which means limiting acceptance to a minimum degree of management quality.

There are two criteria that represent the decision-makers’ preferences, given in the objective functions and restrictions: 1. according to the sum of the weights assigned to the attributes, there will be a scale of scores that will show the degree of improvement in management expected with that alternative; and 2. the sum of the decision variables should be at least 3, which means imposing that no alternative with a score of zero in the decision variables will be accepted, corresponding to a minimum degree of acceptable management quality.

Once the parameters defined in tables 1 to 3 are used, along with the objective function and constraints of the model, it is possible to calculate the feasible outcomes and the respective optimal solutions ( Figure 3), given that the budget constraint (13) will be found in the EE model using the TCO.

In Figure 3, each vector ${\overrightarrow{Z}}^a=\left(Z_x^a,Z_y^a\right)$ is the pair of decision variables chosen from Xª and Yª of alternative A, which is the set of management indicators that make up the fleet management preferred by the decisionmaker.

Since it was decided that the vector should be greater than or equal to 3 (the sum of the management grades revealed in the sequential indicators of the vector), ${\overrightarrow{Z}}^a\ge 3$, any two or more sets of decision variables that have as sum 3, the decision-maker will be indifferent, (2, 1) ~ (1, 2), Z ₁ ~ Z ₂, because they lie on the indifference curve imposed by the constraint (13). A vector ${\overrightarrow{Z}}^a$ with a sum above three will be preferable to one with a sum 3, ${\overrightarrow{Z}}^a=3$ therefore, (2, 2) ≻ (1, 2), meaning that the decision-maker strictly prefers Z ₃ = (2, 2) to Z ₂ = (1, 2).

According to the assumptions, since the costs of each decision variable increase proportionally to the quality grade of the respective item, there will be an increase in the total cost Δ p _iz _i which can overcome the constraint that is the cost (price) limit L _e. This L _e limit, given in Equation (13), is the current cost of maintaining one’s fleet. The central idea is that, in order to change from own to leased fleet, besides the management improvement criteria, costs will not increase. The idea is to focus on efficiency and increase the quality and quantity of services without increasing costs. In (14), it is stated that variations in Δ z _i will cause changes in costs Δ p _iz _i .

The curve ( Figure 3 – objectives space), formalized by restriction (13) of the cost limit L _e, has a negative slope, indicating that whenever the decisionmaker (government) gives up a certain degree of quality in a management indicator X, it will be necessary to compensate with a certain degree of improvement in another management indicator Y. For example, for them to accept reducing the liquidity of the reserve car, they must increase the fleet’s adequacy capacity.

The problem exposed here has multiple objectives, translated into the vector ${\overrightarrow{Z}}^a$ with indicators and their acceptable management degrees that are in the set of optimal solutions, called the Pareto-optimal, or non-dominated, frontier. The objective functions amount to wanting concomitantly a better management with Z _xª = fleet maintenance and Z _yª. In addition, the manager is challenged by the restriction that the management change cannot increase current costs. In this context, Zª = (Z _xª, Z _yª) is the maximization of the management function, given the criteria translated into the indicators.

In Figure 3, the objectives space is plotted with the functions that aim to maximize the fleet management model, given the possibility of the decision on relevant management aspects, defined a priori . The objective functions (1) and (2) map the feasible points in the decision space. The cost parameter is the cost of own fleet, so the change to leased fleet has the restriction equivalent to the costs of remaining with the own fleet of vehicles. cles. The function Zª maps the decision space in search of its images, given its constraints, creating Z* as the feasible objectives space.

Given restriction (11), the two vectors $\overrightarrow{z_1}$ and $\overrightarrow{z_2}$, plotted in Figure 2, have by the decision-maker indifference in preference. At the point (2, 2), there is a bundle with attributes of the degree of change management better because it is above the constraint. However, for outsourced companies to work at this point, the cost would be higher, causing a prohibitive price (above the limit price). At the point (1, 1), there is a bundle with attributes of a lower degree of change, which would cause outsourcers, given lower requirements, to work with a lower cost, leading to a price below the limit price.

A vector $\overrightarrow{z_i}$ with a sum of attributes above 3, (2, 2) ≻ (1, 2), will be preferable over one with a sum $\overrightarrow{z_i}=3$, but, as the costs p _iz _i of each decision variable increases proportionally to the degree of quality, there may be an increase in total cost that can overcome the restriction imposed by the price limit L _e plotted in Figure 2. As for vectors with equal sums, such as (2, 1) ~ (1, 2), Z ₁ ~ Z ₂, the decision-maker will not have a preference. Although (2, 2) may be preferred, it is not possible given the restrictions and the possibility of higher costs, so the vectors $\overrightarrow{z_1}$ and $\overrightarrow{z_2}$ have optimal solutions. Since only one solution is chosen, the set of tie-breaking criteria, translated into the bidding notice, will consider the casting vote.

Given the prices p _x and p _y, which are equivalent to the companies’ revealed costs to make the items available, as presented in Table 1, the alternative (Z _xª, Z _yª has revealed preference. The management indicator degree bundles of the two vectors $\overrightarrow{z_1}$ and $\overrightarrow{z_2}$, given the constraint, revealed a preference for the bundle $\overrightarrow{z_4}$, which could be chosen.

Pareto efficient situation is the frontier line bounded by the restriction; along with this line one has the situation of guaranteed efficiency in fleet management change. Since the acceptable sum of the criteria scores is above 2 and the score function is in ℤ+ (non-negative integers), it is a multiobjective integer linear programming problem.

In the pursuit of the equilibrium price or limit (Le), it is necessary to evaluate the projects or investments in question. It is known that the principle of efficiency should lead to a decision that maximizes the cost-benefit ratio. The efficiency of the means, with a significant reduction in waste, should contribute to increasing social benefits with the same amount of resources, without increasing costs.

Thus, the evaluation by the TCO provides metrics to measure these costs. TCO involves life cycle costing, evaluating the “zero margin” price and assessing the total costs involved ( Ellram & Siferd, 1998).

The projects are not mutually exclusive, that is, the option for one does not necessarily cancel the other. There is the option of using a mix between the two options, 60% of option A (own fleet) and 40% of option B (leased fleet) or another ratio. One must be careful when analyzing exclusively the cost, taking the decision to the lowest cost option, neglecting the focus when it comes to a public good, which is the service to society.

The output flows are clear, deterministic, since they are related to the various expenses involved in the options of own or leased fleets and must be measured in the product’s life cycle. The benefits, in turn, are difficult to measure, since they do not involve inflows of financial resources, but the satisfaction of the population with a service. At this point, one can compare the cash flow outflows by the net present value (NPV) and, then, weigh them with a comparative analysis of the qualitative benefits among the options, such as greater agility in the decision and fleet flexibility, as defined in Table 1.

The economic life of a good is characterized by the optimal point of substitution in which the cost is minimal. In this study, the optimal point of substitution is given, that is, it is an exogenous variable. This happens because the historical data analysis of the Ceará’s state police shows that, after two years, the several costs involved in the own fleet grow exponentially, so 24 months is the time of use. Equation (18) defines the NPV.

The NPV equation:

in which: I is the initial investment; R _Lt is the expected net returns; t is the project review deadline; k is the cost of capital defined by the discount rate; and Rs _t is the residual value of the project at time t.

Once expenditures are analyzed, and since the benefits are incommensurable in monetary terms, Rs _t is taken to zero in the NPV calculation. The benefits must be measured and weighted in the basic model assumptions in a qualitative way, which is done by multiobjective and multicriteria analyses.

Considering R _Lt as the expected net return flows (revenue minus costs), when you take Rs _t to zero, the costs (and expenses) incurred with the own fleet (C _t) and NPV becomes present value of own vehicle (VP _P), as per Equation (19):

As part of the expenses incurred during the maintenance period of the fleet itself, straight-line depreciation is defined from a lifespan of N periods (years, months). Depreciation is the process through which investments made in assets necessary for operation are transformed into costs or expenses, then:

in which:

1. P = price of new vehicle;

   N = service life in years;

   T = period (year, month etc.) of depreciation calculation;

   C _dT = depreciation coefficient for period T;

   D _T = depreciation that the vehicle will suffer in period T;

   DT _T = total depreciation that the vehicle will suffer in its useful life up to period T.

The residual value of the project at time T (R _st) is calculated as follows:

The market, besides the depreciated value of the vehicle, buys it with a haircut, (Dg _T), due to the perception that police cars deteriorate and that the final price of use, already discounting the depreciation, cannot translate the state of the car, leading to a haircut on this price.

It is possible to calculate the total haircut (Dg _T) that the vehicle will suffer in its standardized lifespan until the T. The Txg _mT is the average haircut rate at the end of the standardized lifespan, perceived as compatible by the market. From the Txg _mT, the basic haircut factor in period T (Fbg _T) is calculated. This factor will accumulate in an exponentially increasing haircut until period T, assuming that the public safety car has an increasing deterioration, therefore, an increasing haircut. Accumulating the factor (Fbg _T) over time, we reach the accumulated rate Txg _T, which is the haircut rate (percent) until period T. When T = N, one has Txg _mT = Txg _T. Equations (26) and (27) below demonstrate these definitions:

Once the haircut rate is defined, the total haircut that the vehicle will suffer in its lifespan up to period T(Dg _T) can be calculated.

in which:

1. Rs _T = residual value at the end of period T;

   Fbg _T = basic haircut factor in period T;

   Txg _T = haircut rate up to period T;

   Txg _m = average haircut rate at lifespan end N;

   Dg _T = total haircut that the vehicle will suffer in its useful life up to period T.

Incorporating the haircut (DgT) in Equation (19), it is possible to reach the definitive VP _P equation, as shown in equations (30) and (31):

C _T incorporates costs and expenses inherent to the own fleet, and it may include:

in which:

1. C _T = costs and expenses incurred with the own fleet until period T;

   D _T = depreciation that the own fleet will suffer until period T;

   TI _T = fees and taxes that the own fleet will suffer until period T;

   Sg _T = insurance with the own fleet until period T;

   Mn _T = maintenance (tires, oil, parts etc.) the own fleet will undergo up to period T;

   Vr _T = cost of reserve vehicles for replacement availability until period T; and

   Cad _T = costs and administrative expenses that the own fleet will have until period T.

The VP _P incorporates all the costs of the lifespan of the product of the decision, it is then the present value of a series of payments of expenses made during the use of own fleet vehicles, discounted by a discount rate k.

The disbursements made in the project period refer not only to cost, but to outlay costs, expenses, or investments, and the total effective expenses of period T (Ge _T) variable GeT shows these disbursements brought to period T as an average per period, using the capital recovery factor (CRF) – in this case, it can be called cost of capital recovery (CCR). This can be considered the amount spent per period for the use of the own fleet. GeT is the same concept as the uniform equivalent annual cost (Ueac) for period T.

The difference between the leased fleet and the own fleet is the absence of expense variables that will not be present in the former. In the leased fleet, the lessee receives from another party (the lessor) a good or a service, by means of a leasing contract, being obliged to pay the adjusted price. Hence, the present value of the leased vehicle (VP _L) is the present value of a series of payments related to the leasing contract established during the use of the leased fleet vehicles, discounted by a discount rate k, which can be the opportunity cost.

in which:

1. VP _L = present value of the leased vehicle;

   L _T = lease price per period T (L _T = L for installment, fixed payment);

   T = period (year, month etc.) of the lease;

   K = the cost of capital defined by the discount rate.

There will be an equilibrium situation when the present value of the disbursements of both options is equal – equilibrium in the sense that, if the values are equal, financially it would not make a difference to choose one or the other. In this case, the decision would be based on the perception of which option would bring more social returns and better levels of management of the process.

Even though not measurable quantitatively, it is possible to qualitatively measure the social and management benefits by the variables: satisfaction of police officers and the community, response time, larger time of use, fleet usability, assurance that the cars will always be available, among others. Equalizing the equations, one has:

For a fixed L lease installment, there is equilibrium:

in which:

1. VP _P = present value – own vehicle;

   VP _L = present value – leased vehicle;

   L _T = lease price per period T – monthly lease or other period T;

   L _e = balance lease amount;

   K = opportunity cost or discount rate;

   Ge _T = total effective expenses up to period T;

   Rs _T = residual value at the end of period T;

   Dg _T = total haircut that the vehicle will suffer in its useful life up to period T.

The L _e shows the result of the cost-benefit analysis between own or leased fleets. For values of L _T > L _e > VP _p, there will be no financial advantage in choosing the leased fleet. For values of L _T < L _e < VP _p, it is financially advantageous to choose the leased fleet.

The limit equilibrium price (PL _T) is the price that equalizes the costs between the own and leased fleet, according to Equation (43). It is also called a prohibitive price, since, after this point, it would bring losses as it would exceed the costs of the own fleet.

The safety margin was defined as an insurance against adversities and unexpected situations, such as breach of contracts. The parameter used to define the margin was also the historical percentage of unserviceable cars. With a percentage over the PL _T, the state would have a margin of safety for the risks involved. The optimal price would be the limit price minus the safety margin.

in which:

1. PL _T = limit equilibrium price;

   PO _T = optimum equilibrium price;

   MS _T = safety margin value;

   ms = unit percentage of the safety margin;

   L _e = balance lease amount.

Correspondence concerning this article should be addressed to Régis F. Dantas, Avenida Barão de Studart, 598, Meireles, Fortaleza, Ceará, Brazil, ZIP code 60120-000. Email: regis.dantas@uol.com.br