Solving DEA models in a single optimization stage: Can the non-Archimedean infinitesimal be replaced by a small finite epsilon?

Podinovski, Victor V.; Bouzdine‐Chameeva, Tatiana

doi:10.1016/j.ejor.2016.09.022

Cited by 25 publications

(10 citation statements)

References 18 publications

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“…

\begin{matrix} lefttrue \\ \max p_{k} = \sum_{r = 1}^{s} y_{r k} w_{r} \\ s . t . \\ \begin{matrix} p_{j} = \sum_{r = 1}^{s} y_{r j} w_{r} \leq 1, & j = 1, \dots, n \\ w_{r} \geq ε, & r = 1, \dots, s \end{matrix} \end{matrix}

where,

ε

is the non‐Archimedean parameter added to the model to obtain strictly positive weights. Following Mehrabian et al, Amin and Toloo, and Podinovski and Bouzdine‐Chameeva, a small positive number can be replaced instead of this parameter in model 1. Podinovski and Bouzdine‐Chameeva have discussed the replacement of a small number instead of the non‐Archimedean parameter in the DEA models.…”

Section: Measuring Prosperity Using Deamentioning

confidence: 99%

“…Following Mehrabian et al, Amin and Toloo, and Podinovski and Bouzdine‐Chameeva, a small positive number can be replaced instead of this parameter in model 1. Podinovski and Bouzdine‐Chameeva have discussed the replacement of a small number instead of the non‐Archimedean parameter in the DEA models. As shown in Mehrabian et al, the maximum value of the non‐Archimedean parameter can be obtained by solving the following LP model:

\begin{matrix} lefttrue \\ ε_{\max}^{⁎} = \max ε \\ s . t . \\ \begin{matrix} \sum_{r = 1}^{s} y_{r j} w_{r} \leq 1, & j = 1, \dots, n \\ w_{r} - ε \geq 0, & r = 1, \dots, s \end{matrix} \end{matrix}

…”

Section: Measuring Prosperity Using Deamentioning

confidence: 99%

See 1 more Smart Citation

Measuring global prosperity using data envelopment analysis and OWA operator

Amin

Siddiq

2019

Int J Intell Syst

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Prosperity is one of the key economic indicators of a nation's success. The measure of a country's true prosperity is best achieved by considering a set of criteria and identifying the optimal weights associated with each criterion. This study introduces a novel method for measuring global prosperity by employing a combination of variables that characterize economic wealth and social wellbeing using data envelopment analysis (DEA) and ordered weighted averaging (OWA) operator. It extends the existing global prosperity assessment approach proposed by the Legatum Institute, an international organization that produces a global prosperity index every year. The current Legatum Prosperity Index is obtained by averaging a set of distinct variables, but it fails to identify the optimal variable weights for each country. This is a significant drawback that we address in this study. Using DEA, each country can freely assign optimal weights that are most favorable to achieving maximum prosperity. It provides a flexible and competitive environment in which all countries can present their strengths, thereby creating a level playing field. This study also uses multilevel DEA efficiency frontiers for classifying countries into different groups based on their levels of prosperity score. Additionally, we apply the OWA operator to distinguish further between the countries within each cluster.

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“…

\begin{matrix} lefttrue \\ \max p_{k} = \sum_{r = 1}^{s} y_{r k} w_{r} \\ s . t . \\ \begin{matrix} p_{j} = \sum_{r = 1}^{s} y_{r j} w_{r} \leq 1, & j = 1, \dots, n \\ w_{r} \geq ε, & r = 1, \dots, s \end{matrix} \end{matrix}

where,

ε

Section: Measuring Prosperity Using Deamentioning

confidence: 99%

\begin{matrix} lefttrue \\ ε_{\max}^{⁎} = \max ε \\ s . t . \\ \begin{matrix} \sum_{r = 1}^{s} y_{r j} w_{r} \leq 1, & j = 1, \dots, n \\ w_{r} - ε \geq 0, & r = 1, \dots, s \end{matrix} \end{matrix}

…”

Section: Measuring Prosperity Using Deamentioning

confidence: 99%

Measuring global prosperity using data envelopment analysis and OWA operator

Amin

Siddiq

2019

Int J Intell Syst

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“…The best way to do this is using a lexicographic optimization approach. Alternatively, although in that case care must be taken to select a sufficiently small value for the non‐Archimedean infinitesimal ε (see, e.g., Podinovski and Bouzdine‐Chameeva, ), both phases can be integrated in a single LP as

\begin{matrix} left \underset{λ, β, s^{x}, s^{y}, \hat{x}, \hat{y}}{normalMax} β + ε \cdot (\sum_{i} \frac{s_{i}^{x}}{x_{i 0}} + \sum_{k} \frac{s_{k}^{y}}{y_{k 0}}) \\ left s . t . \\ left \sum_{j} λ_{j} x_{i j} \leq {\hat{x}}_{i} = x_{i 0} - β \cdot g_{i}^{x} - s_{i}^{x} \forall i \\ left \sum_{j} λ_{j} y_{k j} \geq {\hat{y}}_{k} = y_{k 0} + β \cdot g_{k}^{y} + s_{k}^{y} \forall k . \\ left \sum_{j} λ_{j} = 1 \\ left λ_{j} \geq 0 \forall j β normalfree s_{i}^{x} \geq 0 \forall i s_{k}^{y} \geq 0 \forall k \\ left {\hat{x}}_{i} \geq 0 \forall i {\hat{x}}_{i} normalinteger \forall i \in I^{int} \\ left {\hat{y}}_{k} \geq 0 \forall k {\hat{y}}_{k} normalinteger \end{matrix}

…”

Section: Proposed Mem Approachmentioning

confidence: 99%

Efficiency assessment using a multidirectional DDF approach

Lozano

Soltani

2018

Int Trans Operational Res

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This paper proposes a new efficiency measure that aims at capturing the overall distance to the efficient frontier and not the distance corresponding to a specific projection, as most oriented and non‐oriented methods compute. The proposed approach uses a grid‐based method to sample all possible improvement directions from a given operating point and for each direction it computes a directional efficiency score solving a directional distance function model. This provides information about the distribution of the distance to the efficient frontier. Therefore, the proposed multidirectional efficiency measure does not provide a single efficiency score but a distribution of efficiency scores. The minimum, the median, the Q1 and Q3 quartiles and the maximum can be singled out as representatives of that distribution. The latter corresponds to the closest among the computed efficient targets. In addition to the efficiency score distribution, the associated efficient targets are also provided. The proposed approach can take into account integer and nondiscretionary variables and a preference structure. An application of the method to assess the efficiency of material recovery facilities is presented.

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“…where z 0 is the dual variable associated with the equality constraint (10), λ j are the dual variables associated with constraints (11), s + is the dual variable associated with the output weight constraint (12) and s − i (for i = 1, 2, 3) are the dual variables associated with the input weight constraints (13).…”

Section: An Output Oriented Dea Model For Mutual Fundsmentioning

confidence: 99%

“…It is known that the presence in programs (14)-(21) of the non-Archimedean infinitesimal ε may cause computational problems when, as it is often done in commercial softwares, it is replaced by a small positive real number (for instance 10 −6 ); see [10,11]. In order to overcome these problems, it is possible to use a special two-phase procedure that avoids using the non-Archimedean infinitesimal ε by splitting the process of efficiency evaluation in two subsequent linear programs (see [12]); the two-phase procedure has the advantage of reducing the computational drawbacks while providing the same efficiency characterization (see [8] (pp.…”

Section: An Output Oriented Dea Model For Mutual Fundsmentioning

confidence: 99%

Introducing Weights Restrictions in Data Envelopment Analysis Models for Mutual Funds

Basso

Funari

2018

Mathematics

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Data envelopment analysis has been applied in a number of papers to measure the performance of mutual funds, besides a great many applications on the more diverse fields of performance evaluation. The data envelopment analysis models proposed in the mutual funds literature do not generally set restrictions on the weights assigned to the input and output variables. In this paper, we study the effects of the introduction of different weight restrictions on the results of the performance evaluation of mutual funds. In addition, we provide a unified matrix representation for three widely used approaches on weight restrictions: virtual weight restrictions with constraints on all decision-making units (DMUs) (on all funds); virtual weight restrictions with constraints only on the target unit; assurance regions. Using the unified matrix representation of the weights constraints, we formulate the data envelopment analysis (DEA ) efficiency model and express the efficient frontier in a unified way for the different weight restrictions considered. We investigate the effects of the different weight restrictions on the performance evaluation by means of an empirical application on a set of European mutual funds. Moreover, we study the behaviour of the fund performance scores as the restrictions on the weights become increasingly strict.

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Solving DEA models in a single optimization stage: Can the non-Archimedean infinitesimal be replaced by a small finite epsilon?

Cited by 25 publications

References 18 publications

Measuring global prosperity using data envelopment analysis and OWA operator

Measuring global prosperity using data envelopment analysis and OWA operator

Efficiency assessment using a multidirectional DDF approach

Introducing Weights Restrictions in Data Envelopment Analysis Models for Mutual Funds

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