ABSTRACT
Parametric DEA models presume a functional form for the efficiency frontier and are used for resource redistribution. Parabolic DEA is a particular case of these models used to redistribute a single input in Variable Returns to Scale scenarios. In this study, we extend this model to redistribute multiple inputs simultaneously, ensuring that all DMUs will become extreme efficient after redistribution maintaining their outputs. The proposed model is a single multi-objective Linear Programming Problem (PPL) which provides a single optimal solution. To solve this model, two approaches are used, the Weighted Sum of the Objective Functions and the Separation of Variables. Two numerical examples considering single and multiple outputs are used and the results obtained are identical for the two approaches.
Keywords:
parabolic DEA; resources redistribution; variable returns to scale
1 INTRODUCTION
Classic Data Envelopment Analysis (DEA) models (Charnes et al., 1978CHARNES A, COOPER WW & RHODES E. 1978. Measuring the efficiency of decision-making units. European Journal of Operational Research, 2(6): 429-444.; Banker et al., 1984BANKER RD, CHARNES A & COOPER WW. 1984. Some models for estimating technical scale inefficiencies in data envelopment analysis. Management Science, 30(9): 1078-1092.) are tools used for efficiency analysis of a set of productive units, the so-called Decision Making Units (DMUs). Dai et al. (2016DAI Q, LI Y & LIANG L. 2016. Allocating Fixed Costs with Considering the Return to Scale: A DEA Approach. Journal of Systems Science and Complexity, 29(5): 1320-1341.) have stated that issues regarding resources fair allocation or redistribution are one of the main uses of DEA.
In classical models, an inefficient DMU would have freedom of production reaching the efficiency frontier by modifying its inputs or outputs without affecting other DMUs. However, this situation is not feasible in realities of competition or cooperation among DMUs and in limited resources environment, where it is undesirable or impossible to change the total sum of some input or output. Thus, one DMU receiving more resources would result in losses for others.
Cook and Kress (1999COOK WD & KRESS M. 1999. Characterizing an equitable allocation of shared costs: A DEA approach. European Journal of Operational Research, 119(3): 652-661.) have introduced the use of DEA output oriented to allocate shared fixed costs among DMUs in constant returns to scale (CRS) scenarios in an efficiency invariance approach. Following this line, Jahanshahloo et al. (2004JAHANSHAHLOO GR, LOTFI FH, SHOJA N & SANEI M. 2004. An alternative approach for equitable allocation of shared costs by using DEA. Applied Mathematics and Computation, 153(1): 267-274.) have presented an alternative model for resource distribution in constant or variable returns to scale scenarios (CRS and VRS) with no need of Linear Programming Problem resolution. Cook and Zhu (2005COOK WD & ZHU J. 2005. Allocation of shared costs among decision making units: a DEA approach. Computers & Operations Research, 32(8): 2171-2178.) have extended the model of Cook and Kress (1999COOK WD & KRESS M. 1999. Characterizing an equitable allocation of shared costs: A DEA approach. European Journal of Operational Research, 119(3): 652-661.) for cost allocation in an input-oriented CRS scenario. They have also suggested the initial idea for developing the VRS model.
Beasley (2003BEASLEY J. 2003. Allocating fixed costs and resources via data envelopment analysis. European Journal of Operational Research, 147(1): 198-216.) have presented a five-stage model for maximizing DMUs efficiency by resources allocation. However, as pointed by Milioni and Bianca Alves (2013MILIONI AZ & BIANCA ALVES L. 2013. Ten years of research on parametric data envelopment analysis. Pesquisa Operacional, 33(1): 89-104.), this model presents a complex formulation in which a phase requires a nonlinear optimization solution. Lins et al. (2003LINS MPE, GOMES EG & SOARES DE MELLO JCCB. 2003. Olympic ranking based on a zero sum gains DEA model. European Journal of Operational Research, 148(2): 312-322.) have developed the Zero Sum Gains DEA model (ZSG-DEA) to set targets for countries at the Olympic Games considering the total sum of medals fixed. Moreover, Fonseca et al (2010FONSECA ABM, SOARES DE MELLO JCCB, GOMES EG & ANGULO-MEZA L. 2010. Uniformization of frontiers in non-radial ZSG-DEA models: An application to airport revenues. Pesquisa Operacional, 30(1): 175-193.) have introduced a ZSG-DEA model with weight restrictions. Recently, Bouzidis and Karagiannis (2022BOUZIDIS T & KARAGIANNIS G. 2022. Extending the zero-sum gains data envelopment analysis model. Journal of Productivity Analysis.) have extended de ZSG-DEA model to the case of a reverse (undesirable) output. However, most of its uses have been for fixed sum variables redistribution. Similar to the Beasley model, DEA-GSZ does not assure that DMUs will become extreme efficient after redistribution.
Unlike classical DEA models, the so-called Parametric DEA model presumes a functional form for shaping the efficiency frontier and its main use is fixed sum variables redistribution. After reallocation, the efficiency frontier takes the shape of a smooth functional form previously defined by the decision maker. Parametric DEA models provide smoothed efficient frontiers. Other kind of smooth frontiers, although with different goals, are studied in Soares de Mello et al (2002SOARES DE MELLO JCCB, LINS MPE & GOMES EG. 2002. Construction of a smoothed DEA frontier. Pesquisa Operacional, 22(2): 183-201.), Nacif et al. (2009NACIF FB, SOARES DE MELLO JCCB & ANGULO-MEZA L. 2009. Choosing weights in optimal solutions for DEA-BCC models by means of a N-dimensional smooth frontier. Pesquisa Operacional, 29(3): 623-642.), Pereira e Soares de Mello (2015PEREIRA ER & SOARES DE MELLO JCCB. 2015. Uso da suavização da fronteira na determinação de pesos únicos em modelos DEA CCR. Production, 25(3): 585-597.), Brandão and Soares de Mello (2017BRANDÃO LC & SOARES DE MELLO JCCB. 2017. Improvements to Smooth Data Envelopment Analysis. RAIRO - Operations Research, 51(1): 157-171.) and Brandão et al. (2020BRANDÃO LC, SOARES DE MELLO JCCB & DEL-VECCHIO RR. 2020. A linear model for smooth DEA BCC frontiers. Computers and Industrial Engineering, 140.). Smoothed DEA frontiers do not result in multiple optimal weight for efficient DMUs and also do not present indeterminacy on the scale variation (increasing, decreasing or constant) in VRS models (Benicio and Soares de Mello, 2015BENICIO J, & SOARES DE MELLO JCCB. 2015. Productivity Analysis and Variable Returns of Scale: DEA Efficiency Frontier Interpretation. Procedia Computer Science, 55: 341-349.; Benicio and Soares de Mello, 2019BENICIO J, & SOARES DE MELLO JCCB. 2019. Different types of return to scale in DEA. Pesquisa Operacional, 39(2): 245-260.). An important advantage of the parametric DEA is that after the redistribution all DMUs will become extreme efficient, because the parametric frontier contains no weakly efficient regions.
Avellar et al. (2007AVELLAR JG, MILIONI AZ & RABELLO TN. 2007. Spherical frontier DEA model based on a constant sum of inputs. Journal of the Operational Research Society, 58(9): 1246-1251.) have introduced the spherical parametric frontier to redistribute a new input among DMUs. This redistribution considered that the allocation of inputs would be sufficiently fair to make all DMUs efficient and arranged at the spherical frontier. Avellar et al. (2010AVELLAR JG, RABELLO T, RABELLO TN & SIMÃO HP. 2010. On the redistribution of existing inputs using the spherical frontier DEA model. Pesquisa Operacional, 30(1): 1-14.) have studied a global target for the input to be redistributed and, unlike the previous paper, this input already exists in the model. In addition, they have showed that, in some cases, some DMUs would have to increase its inputs to become efficient when using the redistribution approach, which is contrary to classic DEA models assumptions. They have additionally presented a theorem for redistribution using the spherical frontier without increasing any DMU’s inputs. In this specific case, the redistributed input total sum is not constant.
Guedes et al. (2012GUEDES ECC, MILIONI AZ, AVELLAR JVG & SILVA RC. 2012. Adjusted spherical frontier model: Allocating input via parametric DEA. Journal of the Operational Research Society, 63(3): 406-417.) have presented the adjusted spherical frontier model. This new model calculates the spherical frontier using the fraction of the total inputs and outputs of a DMU instead of scalar values. Thus, as they have stated, the model presents results more adherent to the coherence property. Milioni et al. (2011aMILIONI AZ, AVELLAR JG, GOMES EG & SOARES DE MELLO JCCB. 2011. An ellipsoidal frontier model: Allocating input via parametric DEA. European Journal of Operational Research, 209(2): 113-121.) and Bianca Alves et al. (2014BIANCA ALVES L, MILIONI AZ & YOSHIHIRO SOMA N. 2014. Input allocation with the ellipsoidal frontier model. Pesquisa Operacional, 34(1): 39-48.) have studied the input allocation using parameterized efficiency in elliptical geometry. Milioni et al. (2011bMILIONI AZ, AVELLAR J & FREITAS G. 2011. Hyperbolic frontier model: a parametric DEA approach for the distribution of a total fixed output. Journal of the Operational Research Society, 62(6): 1029-1037.) and Silva et al. (2017SILVA RC, MILIONI AZ & TEIXEIRA JE. 2017. The general hyperbolic frontier model: establishing fair output levels via parametric DEA. Journal of the Operational Research Society, 1(1): 946-958.) have introduced the hyperbolic DEA frontier. Milioni and Bianca Alves (2013MILIONI AZ & BIANCA ALVES L. 2013. Ten years of research on parametric data envelopment analysis. Pesquisa Operacional, 33(1): 89-104.) have made a brief overview about related studies on Parametric DEA models.
It is noteworthy that all parametric DEA models previously mentioned only deal with CRS scenarios which generate zero-degree homogeneous functions whenever centred at the origin (Coelli et al., 2005COELLI TJ, RAO DSP, O’DONNELL CJ & BATTESE GE. 2005. An introduction to efficiency and productivity analysis. Springer Science & Business Media.). On the other hand, Silveira et al. (2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.) have developed the parabolic model that respects VRS conditions by parameterizing the frontier using a paraboloid functional form for redistributing one input. Based on this model, Moreira et al (2021MOREIRA LS, SOARES DE MELLO JCCB & ANGULO-MEZA L. 2021. An extension of the parabolic dea frontier to deal with integer variables: A case study in brazilian railway stations. Investigacion Operacional, 42(1): 73-83) developed an extension to take into account integer variables. In this paper, we extend the original Parabolic DEA (Silveira et al., 2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.) to redistribute multiple fixed sum resources in VRS scenarios.
2 PARABOLIC DEA FOR ONE INPUT REDISTRIBUTION REVIEW
Parabolic DEA (Silveira et al., 2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.) is a particular case of parametric models to perform input redistribution in VRS scenarios that may have multiple inputs. However, in order to generate the parabolic frontier, only one of the inputs is allowed to change (redistribute). In such frontier, all DMUs become efficient without changing the total sum of the redistributed input. Thus, all DMUs become extreme efficient without changing neither the outputs nor the other inputs.
The Parabolic DEA model developed by Silveira et al. (2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.) describes the new efficiency frontier by the input being a quadratic function of the output, in one input and one output scenarios. Equation (1) describes the frontier in a set of k DMUs. The new input of a DMU k is represented by x k and the outputs by y k in which k varies from 1 to n. Also, a, b and c are the parabola coefficients.
The model also ensures some inherent properties of a DEA frontier such as convexity, increasing monotonicity, frontier in non-negative quadrants and non-negative values for inputs. Restrictions (2), (3), (4) and (5) represent these conditions respectively.
The objective function, represented by (6), minimizes the difference between the original input of a DMUk (x ok ) and the input obtained after redistribution (x k ). Thus, the new frontier ensures that the computed optimal solution is as close as possible to the original configuration.
To linearize the objective function’s absolute value (or modulus), Silveira et al. (2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.) have adopted the auxiliary variable Mk. For this, they have replaced the objective function (6) by (7) and have included restrictions (8) and (9). In addition, restriction (10) ensures the redistributed input fixed total sum.
Subject to
The Linear Programme (11) describes the parabolic frontier for cases using one input and one output. Unlike classic DEA, the redistribution by parabolic DEA requires a solution of a single Linear Programme.
Subject to
The bi-dimensional model described can be generalised for multiple outputs scenarios. In this case, the frontier previously named as parabolic becomes a paraboloid and is represented by equation (12). Moreover, restrictions to convexity and increasing monotonicity must be adapted to the new outputs as in (13) and (14).
Silveira et al. (2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.) have also generalized the Parabolic DEA model for multiple inputs and outputs scenarios. It is noteworthy that this model only redistributes one input. For this generalization, equation (15) represents the new frontier that will be determined by the redistribution of input x 1. Restrictions (16) and (17) represent the restrictions to frontier’s convexity and increasing monotonicity.
3 PARABOLIC DEA FOR MULTIPLE INPUTS REDISTRIBUTION
In this paper, we extend the Parabolic DEA model developed by Silveira et al. (2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.) to redistribute multiple inputs. For this, as main strategy, we have used the multi-objective approach, that is, one objective function for each input to be redistributed. Equation (18) represents the objective functions.
Similarly to the original model (Silveira et al., 2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.), the auxiliary variable M ik and its restrictions linearize the objective functions.
Subject to
In this approach, we have stated that the efficiency frontier is the intersection of the paraboloid functions that will be determined for each input by the model, as represented in (22).
Therefore, restrictions (23) and (24) assure frontier’s convexity and increased monotonicity. Furthermore, restrictions (25) and (26) state that the independent term of all paraboloids equations and the new inputs must be non-negative.
Combining the objective functions and restrictions presented, the Linear Programme (27) describes the parabolic DEA model for redistributing multiple inputs in one output scenarios.
Subject to
This model can be generalised to redistribute multiple inputs in multiple outputs scenarios by developing the paraboloid surface to consider the new variables, as represented by (28). It is important to stress out that this model do not change outputs.
This generalisation also requires the expansion of the restrictions for convexity and monotonicity to limit all outputs as in (29) and (30).
The Linear Programme (31) describes the parabolic DEA model for redistributing multiple inputs in multiple outputs scenarios.
Subject to
The proposed model respects the characteristics of the original parabolic model developed by Silveira et al. (2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.). It constructs the efficiency frontier as a smooth and rising convex curve, in a shape corresponding to the intersection of two or more paraboloids, which respects VRS conditions. It redistributes multiple input variables keeping their sum constant, making all DMUs extreme efficient and located at the frontier. It is noteworthy that this method results in the absence of restrictions that simultaneously contain all input variables. This condition assures the solution’s uniqueness in multi-objective models.
4 RESOLUTION APPROACHES AND NUMERICAL EXAMPLES
We present two approaches for redistributing multiple inputs using Parabolic DEA. The first one uses scalarizing technique for calculating nondominated solutions (Antunes et al., 2016ANTUNES CH, ALVES MJ & CLÍMACO J. 2016. Multiobjective Linear and Integer Programming. New York: Springer.). That is, the r objective functions are replaced by a single function. In this approach, the single function is represented by a weighted-sum of the objective functions with positive weights λ i , as in (32).
Subject to
This approach requires the variable’s nondimensionalization since it is not possible to sum different measure units. In addition, the variable’s divergence of scale may lead to unfeasible results. This technique consists of dividing each variable by its largest unit with no loss in result since DEA is invariant at scale.
The second approach is the Separation of Variables, which consists in transforming the single Linear Programme in multiple programmes, one for each input. This is possible because there is no restriction that simultaneously consider all input variables, so it is viable to separate each input objective function and its related fixed sum restriction. To illustrate the model, we present a numerical example for a set of five DMUs, one output (y) and two inputs (x 1 and x 2 ) to be redistributed. Table 1 depicts the data set and the classical input-oriented BCC efficiency indexes.
For nondimensionalization, the variables must be divided by its largest values. Thus, we divided variables x 1 , x 2 and y by 25.2, 10 and 5.4, respectively. Table 2 depicts the original data set nondimensionalized.
The objective functions for this example are as shown in (33).
They minimize the auxiliary variable M ik , which represents the difference between the original input i value of DMU k and after redistribution. The variable M ik is related to the module restrictions, represented by (34) in this example.
Equations (35) and (36) represent the restrictions for inputs x 1 and x 2 fixed sum.
The set of paraboloids restrictions represented by (37) for x 1 and (38) for x 2 will define the new efficiency frontier.
Restrictions (39) and (40) ensure the frontier’s positive concavity and its increasing monotonicity, developed from the determination that the second derivative and the first derivative are non- negative.
Using the Weighted Sum scalarizing technique and knowing that Minz λ =Max(−z λ ), the objective functions in (33) become the mono-objective function represented in (41).
The Linear Programme (42) represents the proposed solution for redistributing multiple inputs using Parabolic DEA and Weighted Sum of the Objective Functions resolution. Table 3 depicts the obtained results and the variations used for λ i .
Subject to
Table 3 shows that the three situations have achieved the same results. This indicates the independence of the weighting values for obtaining the results and that both variables represent the same importance for the model. We expected these results since there are no simultaneous restrictions for both input variables in the linear programme (42). This condition allows the Separation of Variables resolution approach.
Using the Separation of Variables approach in the numerical example presented, we obtain two linear programmes, one for input x 1 , described by (43) and the other one for input x 2 , (44). Table 4 depicts the results achieved.
-
• Linear Programme 1 (input x 1 ):
Subject to
-
• Linear Programme 2 (input x 2 ):
Subject to
Table 5 depicts the results achieved by resolutions 1 and 2, Weighted Sum of the Objective Functions and Separation of Variables, respectively, for multiple inputs and one output scenarios. We can verify that both present equal results, which indicates that the input variables represent the same importance for the model.
Figure 1 shows the DMU’s original and redistributed layout, represented respectively by the blue and red dots. We can also visualize the alignment of the DMUs following the parabolic shape by redistributing their inputs without changing outputs. It is interesting to note that even DMUs A and B, considered efficient in the original model, have received resources in order to make them efficient from the perspective of the parabolic DEA. There was no change in DMU E variables.
The paraboloids defined by the model are represented by (45) and (46) for inputs x 1 and x 2 .
Figures 2 and 3 show the paraboloid functions defined for input x 1 and input x 2 on the three- dimensional Cartesian system, respectively.
Figure 4 shows the new DMU’s input configuration determined by the model. Note that they are arranged at the intersection of the two paraboloid functions. Figure 5 simultaneously shows the DMUs following the original and the new input configuration determined by the model, respectively represented by the blue and red dots.
To illustrate this approach in multiple outputs scenarios, we will present another numerical example as shown in Table 6. In this example, we will analyse a set of 5 DMUs in which x 1 and x 2 are the inputs to be redistributed. The outputs are represented by y 1 and y 2 . The difference between numerical examples 1 and 2 is the inclusion of y 2 .
As in Example 1, the variables must be divided by its largest values for nondimensionalization. Table 7 depicts the original data-set nondimensionalized and equation (47) represents the linear programme for this example.
Subject to
Using the Weighted Sum of the Objective Functions resolution, Table 8 depicts the obtained results and the variations used for λ i . As in numerical example 1, the three situations have achieved the same results. This indicates the independence of the weighting values for obtaining the results and that both variables represent the same importance for the model.
The Separation of Variables approach segregates the objective functions and restrictions of each input variable and calculates the PPLs individually. Thus, using this method in numerical example 2 we have achieved the results shown in Table 9.
Equations (48) and (49) represent the paraboloids defined for inputs x 1 and x 2 .
Table 10 depicts the results achieved by resolutions 1 and 2, Weighted Sum of the Objective Functions and Separation of Variables respectively, for multiple inputs and outputs scenarios. As in numerical example 2, both approaches have achieved the same results.
Comparing the results obtained for inputs x 1 and x 2 for both numerical examples, Tables 5 and 10, which have as only difference output y 2 in the second one, it is possible to observe that the results obtained are different. This proves that the output variable’s choice as well as the number of variables considered directly interfere in the result, although the inputs do not influence the results of redistribution between them.
5 FINAL COMMENTS
In this paper, we have extended the parabolic DEA model (Silveira et al., 2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.) to redistribute multiple inputs simultaneously. For this, we have made an overview about parametric DEA models and other DEA-based models for variable redistribution. In addition, we have detailed the original parabolic DEA model (Silveira et al., 2019SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.), which is the main literature source of our research. The solution for multiple inputs redistribution that we have presented in this study guarantees that all DMUs will become extreme efficient and arranged in the new efficiency frontier without changing the outputs. In addition, the model returns the efficiency frontier as a rising convex curve, of a shape corresponding to the intersection of multiple paraboloids defined for each input variable, which meets the VRS conditions.
To define the parabolic efficiency frontier, the model defines a paraboloid function for each input variable, resulting in the absence of restrictions that simultaneously contemplate all the input variables. In addition, an only multi-objective Linear Programming Problem represents this model.
We have presented two resolutions approaches named Weighted Sum of the Objective Functions and Separation of Variables, which were used in two numerical examples in scenarios of one output and multiple outputs. The results achieved were identical for each scenario, showing the independence between the input variables.
We also have observed that the parabolic model for multiple inputs redistribution assures the solution’s uniqueness, unlike classic multi-objective models, which are able to determine only a set of non-dominated solutions. This characteristic is valid regardless the number of inputs analysed, as long as no restrictions simultaneously consider all inputs, that is, as long as the resolution by Separation of Variables is feasible.
In futures studies, we intend to search for a solution that redistributes multiple inputs considering dependence between variables. Moreover, we also intend to study the extension of the model to redistribute multiple outputs. In such case, we believe that the main difficulty will be non-linear nature of the models.
References
- ANTUNES CH, ALVES MJ & CLÍMACO J. 2016. Multiobjective Linear and Integer Programming. New York: Springer.
- AVELLAR JG, MILIONI AZ & RABELLO TN. 2007. Spherical frontier DEA model based on a constant sum of inputs. Journal of the Operational Research Society, 58(9): 1246-1251.
- AVELLAR JG, RABELLO T, RABELLO TN & SIMÃO HP. 2010. On the redistribution of existing inputs using the spherical frontier DEA model. Pesquisa Operacional, 30(1): 1-14.
- BANKER RD, CHARNES A & COOPER WW. 1984. Some models for estimating technical scale inefficiencies in data envelopment analysis. Management Science, 30(9): 1078-1092.
- BEASLEY J. 2003. Allocating fixed costs and resources via data envelopment analysis. European Journal of Operational Research, 147(1): 198-216.
- BENICIO J, & SOARES DE MELLO JCCB. 2015. Productivity Analysis and Variable Returns of Scale: DEA Efficiency Frontier Interpretation. Procedia Computer Science, 55: 341-349.
- BENICIO J, & SOARES DE MELLO JCCB. 2019. Different types of return to scale in DEA. Pesquisa Operacional, 39(2): 245-260.
- BIANCA ALVES L, MILIONI AZ & YOSHIHIRO SOMA N. 2014. Input allocation with the ellipsoidal frontier model. Pesquisa Operacional, 34(1): 39-48.
- BOUZIDIS T & KARAGIANNIS G. 2022. Extending the zero-sum gains data envelopment analysis model. Journal of Productivity Analysis.
- BRANDÃO LC, SOARES DE MELLO JCCB & DEL-VECCHIO RR. 2020. A linear model for smooth DEA BCC frontiers. Computers and Industrial Engineering, 140.
- BRANDÃO LC & SOARES DE MELLO JCCB. 2017. Improvements to Smooth Data Envelopment Analysis. RAIRO - Operations Research, 51(1): 157-171.
- CHARNES A, COOPER WW & RHODES E. 1978. Measuring the efficiency of decision-making units. European Journal of Operational Research, 2(6): 429-444.
- COELLI TJ, RAO DSP, O’DONNELL CJ & BATTESE GE. 2005. An introduction to efficiency and productivity analysis. Springer Science & Business Media.
- COOK WD & KRESS M. 1999. Characterizing an equitable allocation of shared costs: A DEA approach. European Journal of Operational Research, 119(3): 652-661.
- COOK WD & ZHU J. 2005. Allocation of shared costs among decision making units: a DEA approach. Computers & Operations Research, 32(8): 2171-2178.
- DAI Q, LI Y & LIANG L. 2016. Allocating Fixed Costs with Considering the Return to Scale: A DEA Approach. Journal of Systems Science and Complexity, 29(5): 1320-1341.
- FONSECA ABM, SOARES DE MELLO JCCB, GOMES EG & ANGULO-MEZA L. 2010. Uniformization of frontiers in non-radial ZSG-DEA models: An application to airport revenues. Pesquisa Operacional, 30(1): 175-193.
- GUEDES ECC, MILIONI AZ, AVELLAR JVG & SILVA RC. 2012. Adjusted spherical frontier model: Allocating input via parametric DEA. Journal of the Operational Research Society, 63(3): 406-417.
- JAHANSHAHLOO GR, LOTFI FH, SHOJA N & SANEI M. 2004. An alternative approach for equitable allocation of shared costs by using DEA. Applied Mathematics and Computation, 153(1): 267-274.
- LINS MPE, GOMES EG & SOARES DE MELLO JCCB. 2003. Olympic ranking based on a zero sum gains DEA model. European Journal of Operational Research, 148(2): 312-322.
- MILIONI AZ, AVELLAR JG, GOMES EG & SOARES DE MELLO JCCB. 2011. An ellipsoidal frontier model: Allocating input via parametric DEA. European Journal of Operational Research, 209(2): 113-121.
- MILIONI AZ, AVELLAR J & FREITAS G. 2011. Hyperbolic frontier model: a parametric DEA approach for the distribution of a total fixed output. Journal of the Operational Research Society, 62(6): 1029-1037.
- MILIONI AZ & BIANCA ALVES L. 2013. Ten years of research on parametric data envelopment analysis. Pesquisa Operacional, 33(1): 89-104.
- MOREIRA LS, SOARES DE MELLO JCCB & ANGULO-MEZA L. 2021. An extension of the parabolic dea frontier to deal with integer variables: A case study in brazilian railway stations. Investigacion Operacional, 42(1): 73-83
- NACIF FB, SOARES DE MELLO JCCB & ANGULO-MEZA L. 2009. Choosing weights in optimal solutions for DEA-BCC models by means of a N-dimensional smooth frontier. Pesquisa Operacional, 29(3): 623-642.
- PEREIRA ER & SOARES DE MELLO JCCB. 2015. Uso da suavização da fronteira na determinação de pesos únicos em modelos DEA CCR. Production, 25(3): 585-597.
- SOARES DE MELLO JCCB, LINS MPE & GOMES EG. 2002. Construction of a smoothed DEA frontier. Pesquisa Operacional, 22(2): 183-201.
- SILVA RC, MILIONI AZ & TEIXEIRA JE. 2017. The general hyperbolic frontier model: establishing fair output levels via parametric DEA. Journal of the Operational Research Society, 1(1): 946-958.
- SILVEIRA JQ, SOARES DE MELLO JCCB, & ANGULO-MEZA L. 2019. Input redistribution using a parametrica DEA frontier and variable returns to scale: The parabolic efficient frontier. Journal of the Operational Research Society, 70(5): 751-759.
Publication Dates
-
Publication in this collection
21 Aug 2023 -
Date of issue
2023
History
-
Received
19 Sept 2022 -
Accepted
08 Feb 2023