# Algorithms for Multi-Criteria Decision-Making and Efficiency Analysis
Problems

Fuh-Hwa Franklin Liu 111Corresponding author<EMAIL_ADDRESS>Professor
Emeritus, Department of Industrial Engineering and Management, National Yang-
Ming Chiao Tung University, Hsin Chu City, Taiwan 300, Republic of China  Su-
Chuan Shih<EMAIL_ADDRESS>Associate Professor, Department of Business
Administration, Providence University,
Tai Chung City, Taiwan 433, Republic of China

(7 June 2024)

###### Abstract

Multi-criteria decision-making (MCDM) problems involve the evaluation of
alternatives based on various minimization and maximization criteria.
Similarly, efficiency evaluation (EA) methods assess decision-making units
(DMUs) by analyzing their input consumption and output production. MCDM and EA
methods face challenges in managing alternatives and DMUs with varying
capacities across different criteria (inputs and outputs). That leads to
performance assessments often skewed by subjective biases in criteria
weighting. We introduce two innovative scenarios utilizing linear programming-
based Virtual Gap Analysis (VGA) models to address these limitations. This
dual-scenario approach aims to mitigate traditional biases, offering robust
solutions for comprehensively assessing alternatives and DMUs. Our methodology
allows for the influential ranking of alternatives in MCDM problems and
enables each DMU to adjust its input and output ratios to achieve efficiency.
MSC: 90B50; 90C29; 90C08; 91A80; 91B06.

###### keywords:

Decision Support System; Multiple Criteria Decision Making; Efficiency
Analysis.

##  1 Research Background and Objectives

###  1.1 The Background

Decision-makers across various sectors are frequently confronted with complex
multi-criteria decision-making (MCDM) scenarios, necessitating the evaluation
of alternatives against a broad range of criteria. Miettinen surveyed methods
to visualize alternatives in MCDM problems[1]. Sahoo and Goswami[2]
comprehensively reviewed MCDM methodologies, highlighting significant
challenges posed by the subjective assessment of alternatives due to the
heterogeneous capacities of criteria. The efficiency analysis (EA) method,
Data Envelopment Analysis (DEA) is a primary tool employed to address MCDM
challenges[2]. Opricović and Tzeng[3] compared the DEA model and the MCDM
VIKOR method. The terms ”alternative” and ”decision-making unit” (DMU) are
used interchangeably. Conflicting criteria in EA and MCDM methods involve
input and output volumes that must be minimized and maximized. Linear
programming-based DEA models evaluate each DMU’s inefficiency score against
its peers, as detailed by Cooper et al.[4], the DMU under evaluation is
referred to as DMU-o.

Based on a robust theoretical foundation for over fifty years, DEA models[5]
often oversimplify evaluations by assuming homogeneity among DMUs. Each model
employs a set of artificial goal weights to estimate DMU-o’s criteria weights
regardless of input and output configurations, potentially leading to
inaccuracies in operational contexts that reflect the diverse realities of
DMUs. Appendix A provides a concise overview of basic DEA models,
distinguishing them from VGA models. These distinctions are further summarized
in Table 4 in Appendix B.

Despite these limitations, Data Envelopment Analysis (DEA) models have
inspired the development of innovative Virtual Gap Analysis (VGA) scenarios.
VGA, grounded in linear programming theories and operating without
assumptions, provides comprehensive solutions for assessing each DMU-o. These
solutions include:

Adjustment Ratio and Virtual Unit Price: VGA calculates the adjustment ratio
and virtual unit price for each input and output, offering detailed insights
into resource allocation.

Intensities of DMUs: The intensities of DMUs are analyzed to understand their
operational dynamics.

Reference Peers: VGA identifies reference peers for each DMU-o, facilitating
comparative analysis and benchmarking.

Efficiency Score: An efficiency score is assigned to each DMU-o, reflecting
its performance relative to the best practices observed within the dataset.

Sensitivity analysis: VGA allows for sensitivity analysis on the data utilized
in the linear models, enabling an examination of how changes in data affect
the outcomes. This feature is precious for ensuring the robustness and
reliability of the results.

We anticipate that DMU-o will find these solutions achievable in its realistic
practice, enhancing the practical applicability of the analysis. In contrast,
traditional statistics-based EA methods and existing MCDM methods do not offer
such a comprehensive and detailed analysis, highlighting the advantages of the
VGA approach.

###  1.2 Our Algorithms to Solve MCDM and EA Problems

We have developed two scenarios to address MCDM and EA problems. In the first
scenario, we propose the first and second VGA models to estimate the
inefficiency score of each DMU-o, excluding and including the sum of
intensities condition (SIC), which equates the total variable intensities of
DMUs to an SIC scalar. We developed a four-phase procedure, where the first
three phases measure the range of the SIC scalar. In the final phase, DMU-o
selects the SIC scalar to determine achievable adjustment ratios for inputs
and outputs.

Each inefficient DMU has a realistic efficiency score between 0 and 1,
alongside reduction and expansion ratios for inputs and outputs. Each
efficient DMU has zero adjustment ratios and an efficiency score of 1. In the
second scenario, we assess each efficient DMU to estimate its super-
efficiency, which is significantly greater than 1. The super-efficiency
converges to 1 when the efficient DMU adjusts inputs and outputs’ estimated
expansion and reduction ratios.

The third and fourth VGA models employed in the second scenario are
modifications of the first two VGA models. Whether inefficient or efficient,
each DMU comprehensively measures the achievable adjustment ratio and
realistic virtual price per unit of each input and output.

The MCDM decision-making group (DMG) utilizes these two scenarios to select
alternatives with a distinct approach. In the second scenario, the DMG selects
a final SIC scalar for each efficient alternative according to the four-phase
procedure. Under specific subjective considerations in decision-making, the
MCDM DMG employs this systematic method to compare alternatives effectively.

###  1.3 Objectives of the Research

This research introduces innovative linear programming-based VGA models under
two distinct scenarios for evaluating DMU-o. The VGA models are designed to
accommodate the heterogeneity of DMUs, thereby promising a comprehensive
evaluation of DMU-o performance aligned with the systematic virtual unit price
of each criterion. The specific objectives of this study are: [(ii)]

Ensure that DMU-o aligns with reference peers of similar input and output
configurations.

Enable DMU-o to mirror reference peers to precisely calibrate the estimations
of inputs and outputs’ virtual unit prices and adjustment ratios.

Allow DMU-o to select the final SIC scalar within the model to calibrate all
adjustment ratios (AARs) achievable in practice.

For the first scenario, achieve the following outcomes:

  * •

Assign an inefficiency score to DMU-o realistically ranging between 0 and 1.

  * •

Express criteria weaknesses through estimated adjustment ratios, identifying
the weakest criterion as the input or output with the largest adjustment
ratio.

Advance from the first to the second scenario to evaluate each identified
efficient DMU, DMU-o, leading to:

  * •

Assign a super-efficiency score to DMU-o that is realistically larger than 1.

  * •

Observe that as DMU-o adjusts the estimated expansion and reduction ratios of
inputs and outputs, its estimated super-efficiency score deteriorates to 1.

  * •

Express criteria strengths through estimated adjustment ratios, identifying
the strongest criterion as the input or output with the largest adjustment
ratio.

Enable decision-makers in MCDM to effectively compare efficient alternatives
based on their criteria strengths and super-efficiencies.

Section 1.2 introduces the substantial strengths of the first VGA scenario.
The first scenario employs the first and second VGA models, accomplishing
objectives (i)–(iv). These two models are further illustrated in Sections 2.2
and 2.3. The third and fourth VGA models are integrated into the second
scenario to fulfill objectives (i)–(iii) and (v)–(vi). Details of these two
VGA models are offered in Section 3.1 and 3.2.

###  1.4 Early VGA Developments

Hwang[6] contributed to establishing CRS and VRS VGA models, where the former
excludes and includes the SIC, equating to 1. However, issues arose concerning
the virtual goal price for DMU-o and the relationship between these models.
The virtual goal price was subjectively assigned. Past research, including
works by Liu and Liu[7, 8], applied the CRS VGA model in two-phase production
systems. The authors have consulted on unpublished theses that utilized these
models.

###  1.5 Paper Organization

This paper is organized as follows: Section 2 introduces the first and second
VGA models for assessing inefficiency. Section 3 discusses the third and
fourth VGA models for determining super-efficiency. Section 4 describes the
four-phase VGA procedure. Section 5 presents a numerical example to illustrate
the two VGA scenarios. Finally, Section 6 concludes the paper and discusses
the findings. Appendix A points out the reasons why DEA models have incomplete
evaluations. Appendix B compares DEA and VGA models. Appendix C introduces the
computational tools of VGA scenarios.

##  2 First Scenario: Evaluating DMUs’ Inefficiencies

The first scenario assesses the inefficiencies of DMUs by the first and second
VGA models in a four-phase procedure.

###  2.1 Notations

The set $J$, $I$, and $R$, respectively, denotes the $n$ DMUs, $m$ inputs, and
$s$ outputs, where $n\geq 2(m+s)$. The decision matrix, denoted as
$(\textit{X}$, $\textit{Y})$, consists of column vectors $(x_{j},y_{j})$, in
which the elements $(x_{ij},y_{rj})$ represent positive observed volume of
each input-i and output-r of each DMU-j.

The adjustment ratio of each input-i and output-r are decision variables
$Q_{io}$ and $P_{ro}$. In evaluating the inefficient DMU-o by the VGA models
in Section 2, input-i reduces $Q_{io}$ and output-r expands $P_{ro}$.
$\pi_{jo}$ denotes the variable intensity of DMU-j. The variable virtual unit
price for input-i and output-r are $(v_{io},u_{ro})$. Symbols such as $Q_{o}$,
$P_{o}$, $v_{o}$, $u_{o}$, and $\pi_{o}$ represent the vectors’ decision
variables.

The symbol $w_{o}^{c}$ is the scalar unit price corresponding to the SIC.
$\kappa_{o}^{c}$ is the SIC scalar of the TSc model. $\omega_{o}^{a}$ is the
virtual scalar price, $\kappa_{o}^{c}\times w_{o}^{c}$.

We developed a two-step process to determine a systematic unified goal price,
$\tau_{o}$, measured in the virtual currency $. The superscripts # and $\star$
indicate the optimal solutions for a decision variable in the VGA model for
Step I and II.

The symbol of a decision variable with superscript ”$\star$” represents its
optimal solution.

###  2.2 Pure Technical Efficiency (PT) Model-the First VGA Model

This model measures the adjustment ratio and virtual unit price of each input
and output of DMU-o alongside the reference peers’ intensities.

The dual program of the PT model aims to measure the maximum total adjustment
price (TAP) of each DMU-o.

$\delta_{o}^{PT\star}=\max_{Q_{o},P_{o},\pi}\sum_{\forall i\in
I}Q_{io}\tau_{o}+\sum_{\forall r\in R}P_{ro}\tau_{o},\forall o\in J;$ (1)
$s.t.\sum_{\forall j\in J}x_{ij}\pi_{jo}+Q_{io}x_{io}=x_{io},\forall i\in I;$
(2) $-\sum_{\forall j\in J}y_{rj}\pi_{jo}+P_{ro}y_{ro}=-y_{ro},\forall r\in
R;$ (3) $\pi_{o},Q_{o},P_{o}\geq 0.$ (4)

Each adjustment condition for each input (or output), Eqs. (2) and (3),
ensures that an input reduces (or output expands) $Q_{io}$ (or $P_{ro}$) to
the benchmark, $\sum_{\forall j\in J}x_{ij}\pi_{jo}$ (or $\sum_{\forall j\in
J}y_{rj}\pi_{jo})$. Each adjustment condition mirrors the primal program’s
variable virtual price per input-i (or output-r) unit, $v_{io}$ (or $u_{ro}$).
Eq. (1) seeks to maximize the sum of variable AARs multiplied by the specified
goal price. Distinguished by positive intensities, reference peers determine
the AARs under these adjustment conditions. The estimated AARs are realistic
since DMU-o and its reference peers maintain similar input and output
configurations.

The primal program of the PT model aims to measure the total virtual gap
(TVG):

$\Delta_{o}^{PT\star}=\min_{v_{o},u_{o}}\sum_{\forall i\in
I}v_{io}x_{io}-\sum_{\forall r\in R}u_{ro}y_{ro},\forall o\in J;$ (5)
$s.t.\sum_{\forall i\in I}v_{io}x_{ij}-\sum_{\forall r\in R}u_{ro}y_{rj}\geq
0,\forall j\in J;$ (6) $x_{io}v_{io}\geq\tau_{o},\forall i\in I;$ (7)
$y_{ro}u_{ro}\geq\tau_{o},\forall r\in R;$ (8) $v_{o},u_{o}\quad free.$ (9)

Each virtual price condition in Eq. (7) and (8) restricts the virtual price of
input-i and output-r, $x_{io}v_{io}$ and $y_{ro}u_{ro}$, to a unified lower
bound goal price $\tau_{o}$, denominated in virtual currency ($). Each virtual
price condition correlates with the dual program’s variable adjustment ratio,
$Q_{io}$ (or $P_{ro}$).

The virtual gap condition (6) limits each DMU-j to a total virtual gap, the
excess of virtual input and virtual output, $\sum_{\forall i\in
I}v_{io}x_{ij}$ \- $\sum_{\forall r\in R}u_{ro}y_{rj}$, to a minimum of $0.
This condition aligns with the dual program’s dimensionless variable
intensity, $\pi_{jo}$. Efficient reference peers exhibit a zero virtual gap.

DMU-o determines its unified goal price in two steps: (1) The initial goal
price is set at $1, and (2) the second goal price is derived from the optimal
solutions. Using the second goal price, DMU-o is expected to achieve an
estimated virtual input of $1 and ensure that its reference peers share
similar criteria configurations.

In Step I and Step II , substitute $\tau_{o}$ by $\tau_{o}^{\\#}$ = $\$1$ and
$\tau_{o}^{\star}$ = $\$\bar{t}$, respectively. An inefficient DMU-o will
obtain the optimal solutions $\$0\leq\delta_{o}^{PT\\#}=\Delta_{o}^{PT\\#}$,
$\$0\leq\delta_{o}^{PT\star}=\Delta_{o}^{PT\star}\leq\$1$, and
$(Q_{o}^{\star},P_{o}^{\star},\pi^{\star})=(Q_{o}^{\\#},P_{o}^{\\#},\pi^{\\#})$.
Since
$\delta_{o}^{PT\\#}:\delta_{o}^{PT\star}=\tau_{o}^{\\#}:\tau_{o}^{\star}=\$1:\$\bar{t}$,
therefore, $\delta_{o}^{PT\star}$ is a function of $\bar{t}$. Similarly, in
(5), $v_{o}x_{o}$ is a function of $\bar{t}$. Determine the dimensionless
value of $\bar{t}$ to have $v_{o}^{\star}x_{o}$=$\$$1 so that
$\delta_{o}^{PT\star}$ is ensured between $0 and $1. According to
$\tau_{o}^{\\#}:\tau_{o}^{\star}=\$1:\$\bar{t}$, and yields
$v_{o}^{\\#}x_{o}:v_{o}^{\star}x_{o}=\$1:\$\bar{t}$. Use (10) to obtain the
dimensionless value of $\bar{t}$. The solutions of Step II are
$v_{o}^{\star}x_{o}=\$1\textrm{ and }u_{o}^{\star}y_{o}\leq\$1$.

$\bar{t}=\$1/v_{o}^{\\#}x_{o}\textrm{ and }\tau_{o}^{\star}=\$\bar{t}.$ (10)

The optimal solution of (1) as depicted in (11), the total adjustment price
(TAP) of DMU-o in Step II equals $\delta_{xo}^{PT\star}$ plus
$\delta_{yo}^{PT\star}$. (12) expresses the observed multiple inputs and
outputs with distinct measurement units are aggregated into the $[\textit{pure
virtual input (pvInput)},\textit{pure virtual output (pvOutput)}]$ of DMU-j in
Steps I and II, $(\alpha_{j}^{\\#},\beta_{j}^{\\#})$ and
$(\alpha_{j}^{\star},\beta_{j}^{\star})$. The minimized total virtual gap
(TVG) of DMU-o equals to the pvInput minus pvOutput,
$(\alpha_{o}^{\\#}-\beta_{o}^{\\#})$ and
$(\alpha_{o}^{\star}-\beta_{o}^{\star})$ in Steps I and-II .

$\displaystyle\$0\leq\delta_{o}^{PT\\#}=\sum_{\forall i\in
I}Q_{io}^{\\#}\times\$1+\sum_{\forall r\in R}P_{ro}^{\\#}\times\$1;$ (11)
$\displaystyle\$0\leq\delta_{o}^{PT\star}=\sum_{\forall i\in
I}Q_{io}^{\star}\tau_{o}^{\star}+\sum_{\forall r\in
R}P_{ro}^{\star}\tau_{o}^{\star}=\delta_{xo}^{PT\star}+\delta_{yo}^{PT\star}\leq\$1.$

Aggregate inputs and outputs into pvInput and pvOutput.

$\displaystyle\$0\leq(TVG)=(pvInput)-(pvOutput);$ (12)
$\displaystyle\$0\leq\Delta_{j}^{PT\\#}=v_{o}^{\\#}x_{j}-u_{o}^{\\#}y_{j}=\alpha_{j}^{\\#}-\beta_{j}^{\\#};$
$\displaystyle\$0\leq\Delta_{j}^{PT\star}=v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j}=\alpha_{j}^{\star}-\beta_{j}^{\star}\leq\$1.$

The primal program’s objective Eq. (5) is to minimize the virtual gap of
DMU-o, ranging between $0 and $1. At the same time, the dimensionless
inefficiency score is calculated as the ratio of the estimated virtual gap to
the estimated virtual input, ideally ranging between 0 and 1.

######  Theorem 2.1.

The PT model fulfills Condition C1, the estimated PT efficiency between 0 and
1.

Proof. The PT model estimates the technical parameters $v_{o}^{\star}$ and
$u_{o}^{\star}$ to have the minimum virtual gap $\Delta_{o}^{PT\star}$. Which
is converted into the pure technical (PT) inefficiency $F_{o}^{PT\star}$
according to (13), where $E_{o}^{PT\star}$is the efficiency.

$\displaystyle 0\leq
E_{o}^{PT\star}=1-F_{o}^{PT\star}=\beta_{o}^{\star}/\alpha_{o}^{\star}\leq
1;F_{o}^{PT\star}=\Delta_{o}^{PT\star}/\alpha_{o}^{\star}=(\alpha_{o}^{\star}-\beta_{o}^{\star})/\alpha_{o}^{\star}.$
(13)

$E_{o}^{PT\star}$ and $F_{o}^{PT\star}$ are between zero and one, where
$\alpha_{o}^{\star}=\$1$ and $\beta_{o}^{\star}\leq\$1$. $\square$

Certainly, normalizing Step I solutions to attain Step II solutions is a
common practice in mathematical optimization problems. (14) likely represents
this normalization process, where the Step I solutions are adjusted or
transformed to achieve the solutions in Step II. This normalization might
involve scaling or modifying Step I solutions to fulfill specific conditions
required for Step II.

$\displaystyle(\delta_{o}^{PT\star},\Delta_{o}^{PT\star},v_{o}^{\star},u_{o}^{\star})=\bar{t}\times(\delta_{o}^{PT\\#},\Delta_{o}^{PT\\#},v_{o}^{\\#},u_{o}^{\\#}).$
(14)

The symbol $\mathcal{E}_{o}^{PT}$ denotes the set of reference DMUs of DMUo in
the PT evaluations. Each reference DMU-j is efficient, and its estimated
intensity, $\pi_{jo}^{\star}>0$. At the same time, the estimated intensity of
the other inefficient DMUs, $\pi_{jo}^{\star}=0$. DMU-o estimates the
benchmark of each input-i and output-r, $\widehat{x}_{io}^{PT\star}$ and
$\widehat{y}_{ro}^{PT\star}$ via (15). DMU-o mimics reference DMUs with their
estimated intensities, $\pi_{jo}^{\star}$, $\forall j\in\mathcal{E}_{o}^{PT}$.

$\displaystyle\widehat{x}_{io}^{PT\star}=\sum_{\forall
j\in\mathcal{E}_{o}^{PT}}x_{ij}\pi_{jo}^{\star}=x_{io}(1-Q_{io}^{\star}),\forall
i\in I;$ (15) $\displaystyle\widehat{y}_{ro}^{PT\star}=\sum_{\forall
j\in\mathcal{E}_{o}^{PT}}y_{rj}\pi_{jo}^{\star}=y_{ro}(1+P_{ro}^{\star})\forall
r\in R.$

######  Theorem 2.2.

PT model fulfills Condition C2, the adjusted PT efficiency equals 1.

Proof. Assessing DMU-o with the adjusted input-i and output-r,
$\widehat{x}_{io}^{PT\star}$ and $\widehat{y}_{ro}^{PT\star}$, as shown in
(15), will have the PT efficiency $\widehat{E}_{o}^{PT\star}$ equals 1.
$\square$ Let the total of estimated intensities be the first SIC scalar,
$\kappa_{o}^{1}$, computed via (16).

$\sum_{\forall j\in\mathcal{E}_{o}^{PT}}\pi_{jo}^{\star}=\kappa_{o}^{1}.$ (16)

###  2.3 Technical and Scalar Choice (TSc) Model

According to (15), the benchmarks of inputs and outputs are the function of
the intensities of DMUs. One can manipulate the SIC scalar to choose the
proper assessments. Adding (20) to the PT model’s dual program will have the
TSc model’s dual program. The SIC, $\sum_{\forall j\in
J}\pi_{jo}=\kappa_{o}^{c}$, corresponds to the free-in-sign decision variable,
$w_{o}^{c}$. The optimal solutions are affected by the SIC. The virtual goal
price $\tau_{o}^{\star}$ is systematically determined in two steps.

TAP (dual) program of the TSc model:

$\delta_{o}^{TSc\star}=\max_{Q_{o},P_{o}}\sum_{\forall i\in
I}Q_{io}\tau_{o}+\sum_{\forall r\in R}P_{ro}\tau_{o},\forall o\in J;$ (17)
$s.t.\sum_{\forall j\in J}x_{ij}\pi_{jo}+Q_{io}x_{io}=x_{io},\forall i\in I;$
(18) $-\sum_{\forall j\in J}y_{rj}\pi_{jo}+P_{ro}y_{ro}=-y_{ro},\forall r\in
R;$ (19) $\sum_{\forall j\in J}\pi_{jo}=\kappa_{o}^{c};$ (20)
$\pi_{o},Q_{o},P_{o}\geq 0.$ (21)

TVG (primal) program of the TSc model:

$\displaystyle\Delta_{o}^{TSc\star}=\min_{x_{o},v_{o},w_{o}^{c}}\sum_{\forall
i\in I}v_{io}x_{io}$ $\displaystyle-\sum_{\forall r\in
R}u_{ro}y_{ro}+\kappa_{o}^{c}w_{o}^{c},\forall o\in J;$ (22)
$s.t.\sum_{\forall i\in I}v_{io}x_{ij}-\sum_{\forall r\in
R}u_{ro}y_{rj}+1w_{o}^{c}\geq 0,\forall j\in J;$ (23)
$x_{io}v_{io}\geq\tau_{o},\forall i\in I;$ (24)
$y_{ro}u_{ro}\geq\tau_{o},\forall r\in R;$ (25) $v_{o},u_{o},w_{o}^{c}\quad
free.$ (26)

In Steps I and II , substitute $\tau_{o}$ by $\tau_{o}^{\\#}=\$1$ and
$\tau_{o}^{\star}=\$\bar{t}$ will solve the TAP program comprehensively with
the solutions
$(Q_{o}^{\star},P_{o}^{\star},\pi^{\star})=(Q_{o}^{\\#},P_{o}^{\\#},\pi^{\\#})$.
In (22), the elements $(\kappa_{o}^{c}w_{o}^{\\#})$ and
$(\kappa_{o}^{c}w_{o}^{\star})$ are repressed by the symbols
$\omega_{o}^{c\\#}$ and $\omega_{o}^{c\star}$, the  virtual scalar $ (vScalar)
in Steps I and II.

As depicted in (27), $\gamma_{o}$ and $(1-\gamma_{o})$ are the proportions of
the total adjustment prices of inputs and outputs. If $w_{o}^{c\\#}$ (or
$w_{o}^{c\star}$) equals $\$0$, let $\gamma_{o}$ equal 0.5. Partition the
vScalar into two parts to reflect the effects of the SIC on inputs and outputs
of DMU-o. We denote them as _vScalar in inputs (ivScalar)_ and _vScalar in
outputs (ovScalar)_ ; vScalar equals ivScalar plus ovScalar.

$\displaystyle\gamma_{o}:(1-\gamma_{o})=\tau_{o}^{\\#}\times\sum_{\forall i\in
I}Q_{io}^{\\#}:\tau_{o}^{\\#}\times\sum_{\forall r\in
R}P_{ro}^{\\#}=\tau_{o}^{\star}\times\sum_{\forall i\in
I}Q_{io}^{\star}:\tau_{o}^{\star}\times\sum_{\forall r\in R}P_{ro}^{\star}.$
(27)

The minimized TVG of DMU-o in Steps II of (22) is expressed as (28); Step I
has a similar expression. The estimated TVG, $\Delta_{o}^{TSc\star}$ equals
the _gray virtual input (gvInput_ ) minus _gray virtual output (gvOutput_ )
plus vScalar. $\Delta_{o}^{TSc\star}$ equals the affected virtual Input
(avInput) minus affected virtual Output (avOutput).

$\displaystyle\$0\leq\Delta_{o}^{TSc\star}$
$\displaystyle=v_{o}^{\star}x_{o}-u_{o}^{\star}y_{o}+\omega_{o}^{c\star}=[v_{o}^{\star}x_{o}+(1-\gamma_{o})\omega_{o}^{c\star}]-(u_{o}^{\star}y_{o}-\gamma_{o}\omega_{o}^{c\star})$
(28) $\displaystyle=[gvInput+ivScalar]-[gvOutput-ovScalar]$
$\displaystyle=avInput-
avOutput=\alpha_{o}^{c\star}-\beta_{o}^{c\star}\leq\$1;$
$\displaystyle\$0\leq\Delta_{o}^{TSc\\#}$
$\displaystyle=\alpha_{o}^{c\\#}-\beta_{o}^{c\\#}.$

Similarly, the estimated vScalar of Step II in (23), $1w_{o}^{*}$, is
decomposed into two components; Step I solutions has the similar expressions.
As shown in (29), the TVG of each DMU-j is expressed as the gap between the
avInput and avOutput.

$\displaystyle\$0\leq\Delta_{j}^{TSc\star}$
$\displaystyle=v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j}+1w_{o}^{c\star}$ (29)
$\displaystyle=[v_{o}^{\star}x_{j}+(1-\gamma_{o})1w_{o}^{c\star}]-(u_{o}^{\star}y_{j}-\gamma_{o}1w_{o}^{c\star})=\alpha_{j}^{c\star}-\beta_{j}^{c\star},\forall
j\in J.$

Each DMU-j is expressed by the pair of _(avInput, avOutput)_ , the virtual
scales. The symbol $\mathcal{E}_{o}^{TS}$ denotes the set of reference peers
in evaluating DMU-o by the TSc model, where each reference DMU-j has
$\pi_{jo}^{\star}>0$. The other inefficient DMU-j has $\pi_{jo}^{\star}=0$. In
(29), any best DMU-j belongs to $\mathcal{E}_{o}^{TS}$ has
$\alpha_{j}^{c\star}=\beta_{j}^{c\star}$, while the other remaining DMUs have
$\alpha_{j}^{c\star}>\beta_{j}^{c\star}$.

######  Theorem 2.3.

The TSc model fulfills Condition C1, the estimated TSc efficiency score
between 0 and 1.

Proof. Using (30) would have the dimensionless value of $\bar{t}$ by setting
the avInput in Step II $\alpha_{o}^{c\star}$ equals $\$1$. Therefore, the TVG
in Step II , $\Delta_{o}^{TSc\star}$, should be between $0 and $1.

$\displaystyle\$\bar{t}=\$1/\alpha_{o}^{c\\#}=\$1/[v_{o}^{\\#}x_{o}+(1-\gamma_{o})\omega_{o}^{c\\#}]=\tau_{o}^{\star}.$
(30)

Because $\tau_{o}^{\\#}:\tau_{o}^{\star}=\$1:\$\bar{t}$, therefore, if
$\tau_{o}^{\\#}$ = $\$\bar{t}$, then $\alpha_{o}^{c\star}$ equals $1.
According to (28), $\alpha_{o}^{c\star}$ \- $\beta_{o}^{c\star}$ is greater
than $0. Using (31), the estimated virtual gap $\Delta_{o}^{TSc\star}$ is
converted into the maximized TSc inefficiency score, $E_{o}^{TSc\star}$,
should be between 0 and 1, in which the sum of $F_{o}^{TSc\star}$ and
$E_{o}^{TSc\star}$ equals 1.

$\displaystyle 0<F_{o}^{TSc\star}$
$\displaystyle=1-E_{o}^{TSc\star}=\Delta_{o}^{TSc\star}/\alpha_{o}^{c\star}=(\alpha_{o}^{c\star}-\beta_{o}^{c\star})/\alpha_{o}^{c\star}<1;$
(31) $\displaystyle 0<E_{o}^{TSc\star}$
$\displaystyle=\frac{u_{o}^{\star}y_{o}-\gamma_{o}\omega_{o}^{c\star}}{v_{o}^{\star}x_{o}+(1-\gamma_{o})\omega_{o}^{c\star}}=\beta_{o}^{c\star}/\alpha_{o}^{c\star}<1.$

$\square$

Normalizing the solutions of Step I by the dimensionless value $\bar{t}$ would
obtain the solutions of Step II, as shown in (32).

$\displaystyle(\Delta_{o}^{TSc\star},\delta_{o}^{TSc\star},v_{o}^{\star},u_{o}^{\star},w_{o}^{c\star})=\quad\bar{t}\times(\Delta_{o}^{TSc\\#},\delta_{o}^{TSc\\#},v_{o}^{\\#},u_{o}^{\\#},w_{o}^{c\\#}).$
(32)

Use (33) to compute the benchmark of each performance index,
$\widehat{x}_{io}^{TSc\star}$ , and $\widehat{y}_{ro}^{TSc\star}$. DMU-o
imitates the best peers with their estimated intensities $(\pi_{jo}^{\star})$.
DMU-o would become efficient with the benchmarks.

$\displaystyle\widehat{x}_{io}^{TSc\star}$ $\displaystyle=\sum_{\forall
j\in\mathcal{E}_{o}^{TS}}x_{ij}\pi_{jo}^{\star}=x_{io}(1-Q_{io}^{\star}),\forall
i\in I;$ (33) $\displaystyle\widehat{y}_{ro}^{TSc\star}$
$\displaystyle=\sum_{\forall
j\in\mathcal{E}_{o}^{TS}}y_{rj}\pi_{jo}^{\star}=y_{ro}(1+P_{ro}^{\star}),\forall
r\in R.$

######  Theorem 2.4.

The TSc model fulfills Condition C2, the adjusted DMU-o has the TSc efficiency
equals 1.

Proof. Assessing DMU-o with the adjusted input-i and output-r as depicted in
(33), $\widehat{x}_{io}^{TSc\star}$ and $\widehat{y}_{ro}^{TSc\star}$, by the
TSc model will have $\delta_{o}^{TSc\star}=\$0$ and the TSc efficiency
$\widehat{E}_{o}^{TSc\star}=1$. $\square$

###  2.4 Duality Properties

We formulated VGA models based on the theories of linear programming (Dantzig
and Thapa, 1997). When the optimal TVG and TAP of the two VGA models are
equal, as stated in (34), it suggests an equilibrium or balance between the
defined conditions or objectives within these models. This equality could
signify an alignment between the virtual gap and slack price considerations,
indicating a harmonized solution or relationship between the gap and price
metrics.

$\displaystyle\delta_{o}^{PT\star}=\Delta_{o}^{PT\star}\quad\textrm{and}\quad\delta_{o}^{TSc\star}=\Delta_{o}^{TSc\star}.$
(34)

The strong complementary slackness conditions of the two VGA models are shown
in (35) and (36).

$\displaystyle[\sum_{\forall
j\in\mathcal{E}_{o}^{VGA}}x_{ij}\pi_{jo}^{\star}-x_{io}(1-Q_{io}^{\star})]v_{io}^{\star}=\$0,\forall
i\in I.$ (35) $\displaystyle[\sum_{\forall
j\in\mathcal{E}_{o}^{VGA}}(y_{rj}\pi_{jo}^{\star})-y_{ro}(1+P_{ro}^{\star})]u_{ro}^{\star}=\$0,\forall
r\in R.$ (36)

$v_{o}$ and $u_{o}$ are free-in-signs in the TVG programs. Since $(\textit{X,
Y})>0$ and $\tau_{o}^{\star}>\$0$ yields the estimated virtual unit prices,
$v_{o}^{\star}>0,u_{o}^{\star}>0$. Therefore, the braces at the left-hand side
of (35) and (36) equal zero.

For the PT model, each DMU-j has the property as shown in (37). When
$(v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j})=0$, DMU-j belongs to
$\mathcal{E}_{o}^{PT}$ and $\pi_{jo}^{\star}>0$. The _Efficiency Equator_ is
the diagonal line at the origin in Figure 2, that
$(v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j})=\$0,\forall
j\in\mathcal{E}_{o}^{PT}$.

$(v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j})\pi_{jo}^{\star}=\$0,\forall j\in J.$
(37)

For the TSc model, each DMU-j has the property shown in (38). $w_{o}^{\star}$
is the estimated unit price of the SIC scalar, and
$\kappa_{o}^{c}w_{o}^{\star}$ is the vScalar. When
$(v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j}+1w_{o}^{\star})=\$0$, DMU-j belongs to
$\mathcal{E}_{o}^{TS}$ and $\pi_{jo}^{\star}>0$. Otherwise,
$(v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j}+1w_{o}^{\star})>\$0$, DMU-j does not
belong to $\mathcal{E}_{o}^{TS}$ and $\pi_{jo}^{\star}=0$. The Efficiency
Equator is the diagonal line at the origin in Figure 3, that
$(v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j}+1w_{o}^{\star})=\$0,\forall
j\in\mathcal{E}_{o}^{TS}$.

$(v_{o}^{\star}x_{j}-u_{o}^{\star}y_{j}+w_{o}^{\star})\pi_{jo}^{\star}=\$0,\forall
j\in J.$ (38)

(39) depicts the relationships between the estimated decision variables in the
TVG and TAP programs.

$\displaystyle(v_{io}^{\star}x_{io}-\tau_{o}^{\star})Q_{io}^{\star}=\$0,\forall
i\in I;(u_{ro}^{\star}y_{ro}-\tau_{o}^{\star})P_{ro}^{\star}=\$0,\forall r\in
R.$ (39)

In (40), $\sum_{\forall
j\in\mathcal{E}_{o}^{TS}}\pi_{jo}^{\star}-\kappa_{o}^{c}=0$ so that TVG of the
TSc model is decreased, constant, and increased when
$w_{o}^{\star}>\$0,w_{o}^{\star}=\$0,\textrm{ and }w_{o}^{\star}<\$0$,
respectively.

$(\sum_{\forall
j\in\mathcal{E}_{o}^{TS}}\pi_{jo}^{\star}-\kappa_{o}^{c})w_{o}^{\star}=\$0.$
(40)

###  2.5 Post-Analysis of the Optimal Solutions

The optimal solutions of the PT and TSc VGA models can quantify the assessment
items.

####  2.5.1 Virtual Technology Sets

The formulation and definitions of the virtual technology set in the context
of Step II within the PT and TSc models are presented in (41) and (42); Step I
has the similar expressions. In which, the virtual scales of each DMU-j
(pvInput, pvOuput) and (avInput, avOutput) are defined in (12) and (28). The
virtual scales in Step I for each DMU-j represent certain aspects or
characteristics within the model. These sets likely encapsulate specific
parameters, constraints, or variables pertinent to the subsequent steps’
formulation and computation. Virtual scales in Step II might represent refined
or adjusted versions of the initial sets of Step I, potentially incorporating
the outcomes or adjustments derived from the earlier steps.

The transition from Step I to Step II usually involves refining or re-
calibrating the parameters or sets based on the intermediate solutions or
computations from the preceding step. This iterative process often helps
converge toward more accurate or optimal results within the models.

$\displaystyle\Phi_{o}^{PT\star}$
$\displaystyle=\\{(\alpha^{\star},\beta^{\star})\mid(\alpha_{j}^{\star},\beta_{j}^{\star}),\forall
j\in J\\}.$ (41)
$\displaystyle\Phi_{o}^{TSc\star}=\\{(\alpha^{c\star},\beta^{c\star})\mid(\alpha_{j}^{c\star},\beta_{j}^{c\star}),\forall
j\in J$ $\displaystyle\\}.$ (42)

####  2.5.2 Return-to-virtual-scale (RTvS)

(43) and (44) depict the computation steps for DMU-o’s benchmark virtual
scales and affected benchmark virtual scale in the PT and TSc models during
Step II, ( bvInput, bvOutput) = ($\widehat{\alpha}_{o}^{\star}$,
$\widehat{\beta}_{o}^{\star}$) and ( abvInput, abvOutput) =
($\widehat{\alpha}_{o}^{c\star}$, $\widehat{\beta}_{o}^{c\star}$),
respectively. These equations likely involve transforming or deriving
benchmark values based on specific conditions or constraints within the
models.

The expressions for bvInput, bvOutput, abvInput, and abvOutput in Step I are
similar to those in (43) and (44) with the replacement of the superscript
”$\star$” with ”$\\#$.” This step-wise process of deriving and refining
benchmarks could indicate a progressive approach where the initial estimates
from Step I are further modified or calibrated in Step II based on additional
considerations or parameters introduced in the models.

$\widehat{\alpha}_{o}^{\star}=\sum_{\forall i\in
I}\widehat{x}_{io}v_{io}^{\star}\textrm{ and
}\widehat{\beta}_{o}^{\star}=\sum_{\forall r\in
R}\widehat{y}_{ro}u_{ro}^{\star}.$ (43)
$\displaystyle\widehat{\alpha}_{o}^{c\star}=[v_{o}^{\star}\widehat{x}_{o}+(1-\gamma_{o})\omega_{o}^{c\star}];\widehat{\beta}_{o}^{c\star}=(u_{o}^{\star}\widehat{y}_{o}-\gamma_{o}\omega_{o}^{c\star}).$
(44)

(45) and (46) compute $\Xi_{o}^{PT\star}$ and $\Xi_{o}^{TSc\star}$ that based
on the virtual scales. The RTvS values might represent a measure or index that
assesses the efficiency or productivity performance of DMU-o within the
context of these specific models.

$\Xi_{o}^{PT\star}=(\widehat{\beta}_{o}^{\star}/\beta_{o}^{\star})/(\widehat{\alpha}_{o}^{\star}/\alpha_{o}^{\star})=1/E_{o}^{PT\star}.$
(45)
$\Xi_{o}^{TSc\star}=(\widehat{\beta}_{o}^{c\star}/\beta_{o}^{c\star})/(\widehat{\alpha}_{o}^{c\star}/\alpha_{o}^{c\star})=1/E_{o}^{TSc\star}.$
(46)

For the case that $\omega_{o}^{\star}\geq\$0$, $\Xi_{o}^{TSc\star}$ and
$E_{o}^{TSc\star}$ will be decreased and increased as the SIC scalar
$\kappa_{o}^{c}$ rises. Conversely, when $\omega_{o}^{\star}\leq\$0$,
$\Xi_{o}^{TSc\star}$ and $E_{o}^{TSc\star}$ will be decreased and increased,
respectively, as the SIC scalar $\kappa_{o}^{c}$ diminishes. A preference is
given to higher $\Xi_{o}^{TSc\star}$ and $E_{o}^{TSc\star}$. Depending on
whether $\omega_{o}^{\star}\geq\$0$ or $\omega_{o}^{\star}\leq\$0$, DMU-o has
the flexibility to adjust $\kappa_{o}^{c}$ towards its lower or upper bound.
This adjustment aims to achieve the final $\kappa_{o}^{z}$ that encompasses
achievable benchmarks for inputs and outputs. The implications of these
dynamics are further elucidated through the numerical examples and graphical
illustrations presented in Section 5.1, which depict how varying
$\kappa_{o}^{c}$ influences the efficiency and the position of DMU-o relative
to the Efficiency Equator.

####  2.5.3 Interconnections Between Inputs and Outputs Indices

(47) and (48) demonstrate adjustments to the affected virtual prices of an
input, $(v_{io}^{\star}x_{io}+\gamma_{io}^{Q}\omega_{o}^{c\star})$, and an
output, $(u_{ro}^{\star}y_{ro}-\gamma_{ro}^{P}\omega_{o}^{c\star})$,
respectively considering the proportions to the vScalar, represented by
$\gamma_{io}^{Q}$ and $\gamma_{ro}^{P}$.

$\displaystyle\Delta_{o}^{TSc\star}=\sum_{\forall i\in
I}(v_{io}^{\star}x_{io}+\gamma_{io}^{Q}\omega_{o}^{c\star})-\sum_{\forall r\in
R}(u_{ro}^{\star}y_{ro}-\gamma_{ro}^{P}\omega_{o}^{c\star}).$ (47)
$\displaystyle\gamma_{io}^{Q}$
$\displaystyle=(1-\gamma_{o})Q_{io}^{\star}/\sum_{\forall i\in
I}Q_{io}^{\star},\forall i\in I;$ (48) $\displaystyle\gamma_{ro}^{P}$
$\displaystyle=\gamma_{o}P_{ro}^{\star}/\sum_{\forall r\in
R}P_{ro}^{\star},\forall r\in R.$

In (47) and (48), the proportions involving
($v_{io}^{\star}x_{io}+\gamma_{io}^{Q}\omega_{o}^{c\star}$ ) and
($u_{ro}^{\star}y_{ro}-\gamma_{ro}^{P}\omega_{o}^{c\star}$) indeed highlight
the interconnectedness between the input and output indices. These proportions
indicate the adjustments made to the affected virtual prices concerning the
quantities represented by $\gamma_{io}^{Q}$ and $\gamma_{ro}^{P}$, displaying
how changes in one influence the other within the context of the model
variables and constraints.

####  2.5.4 2D Graphic Intuitions

DMU-o can visualize the outcomes of the post-analysis based on the optimal
solutions from the VGA models. For instance, I am using the example dataset in
Table 1. One can identify the third SIC scalar $\kappa_{o}^{3}$ by try and
error. While the $\kappa_{o}^{3}$ may not be within the range of
$\kappa_{o}^{1}$ and $\kappa_{o}^{2}$. The particular $\kappa_{o}^{3}$ within
the TS3 model produces $E_{o}^{PT\star}=E_{o}^{TS3\star}$. Pure technical
efficiency is affected by the SIC. Figure 2 depicts the solutions of PT and
TS3 models. Points Kp, K3, AP3, and O are located at
($\alpha_{o}^{\star},\beta_{o}^{\star}$),
($\alpha_{o}^{3\star},\beta_{o}^{3\star}$),
($[1-\gamma_{o}]\omega_{o}^{3\star},-\gamma_{o}\omega_{o}^{3\star}$), and
(0,0), respectively. Points O, K3, and AP3 at the triangle. The upper-right of
the figure shows the two rectilinear distances between the pair points (Kp,
Tp) and (K3, T3), the virtual gaps estimated by the PT and TS3 models. DMU-o
could comprehend the results between the PT and TS3 models to understand the
effects of the SIC.

Similarly, in Figure 3, Points O, K1, and AP1 of the triangle are solutions of
the TS1 model. The three points O, K2, and AP2 at the triangle are solutions
of the TS2 model. Figure 3 showcases the effects of using different SIC
scalars within a certain interval and helping select a final scalar
$\kappa_{o}^{z}$ for productivity management. In this example, due to
$w_{o}^{\star}>0$, we have $E_{o}^{TS2\star}$ $>$ $E_{o}^{TSz\star}$ $>$
$E_{o}^{TS1\star}$. The TSz model provides the most preferred AARs
($Q_{o}^{TSz\star},P_{o}^{TSz\star}$).

In Figure 3, Points O, APc, and Kc at the triangle which expresses solutions
of the TSc model, where ”c” denotes the choices 1, 2, and 3. (49), (50), and
(51) are the expressions of the slopes for the vectors $\overrightarrow{\rm
O,Kc}$, $\overrightarrow{\rm O,APc}$, and $\overrightarrow{\rm APc,Kc}$,
respectively. The ratio of the scalar prices of inputs to outputs equals the
slope $\bar{m}(\overrightarrow{\rm O,APc})$. Similarly, the ratio of the gray
virtual input to output equals the slope $\bar{m}(\overrightarrow{\rm
APc,Kc})$.

$\displaystyle\bar{m}(\overrightarrow{\rm
O,Kc})=\frac{{\beta_{o}^{c\star}-0}}{{\alpha_{o}^{c\star}-0}}=\frac{{\beta_{o}^{c\star}}}{{\alpha_{o}^{c\star}}}.$
(49) $\displaystyle\bar{m}(\overrightarrow{\rm
O,APc})=\frac{-\gamma_{o}\omega_{o}^{c\star}-0}{(1-\gamma_{o})\omega_{o}^{c\star}-0}=\frac{-\gamma_{o}\omega_{o}^{c\star}}{(1-\gamma_{o})\omega_{o}^{c\star}}.$
(50) $\displaystyle\bar{m}(\overrightarrow{\rm
APc,Kc})=\frac{\beta_{o}^{c\star}-(-\gamma_{o}\omega_{o}^{c\star})}{{\alpha_{o}^{c\star}}-[-(1-\gamma_{o})\omega_{o}^{c\star}]}=\frac{u_{o}^{\star}y_{o}}{v_{o}^{\star}x_{o}}.$
(51)

For instance, in Figure 2, the triangle of the TS3 model has the following
expressions.

$\displaystyle\overrightarrow{\rm O,K3}=\overrightarrow{\rm
O,AP3}+\overrightarrow{\rm
AP3,K3}\iff\frac{\beta_{o}^{3\star}}{\alpha_{o}^{3\star}}=\frac{u_{o}^{\star}y_{o}-\gamma_{o}\omega_{o}^{3\star}}{v_{o}^{\star}x_{o}+(1-\gamma_{o})\omega_{o}^{3\star}}.$
(52)

######  Theorem 2.5.

TSc model fulfills Condition C3.

Proof. As shown in (53), the estimated TSc efficiency score has the following
expressions.

$\displaystyle\overrightarrow{\rm O,Kc}=\overrightarrow{\rm
O,APc}+\overrightarrow{\rm APc,Kc};\quad
E_{o}^{TSc\star}=\bar{m}(\overrightarrow{\rm
O,Kc})=\frac{\beta_{o}^{c\star}}{\alpha_{o}^{c\star}}.$ (53)

The effects of the SIC within the TSc model could be visualized as triangular
with points O, APc, and Kc in the 2D graphical intuition. The slopes
$\bar{m}(\overrightarrow{\rm O,APc})$ and $\bar{m}(\overrightarrow{\rm
APc,Kc})$ are not the scale and technical efficiencies. $\square$

##  3 Second Scenario: Evaluating Efficient DMUs’ Super-Efficiencies

In the first scenario, DMU-o is identified as efficient when
$E_{o}^{sPT\star}$=1. Let efficient DMUs belong to the set Ė. PT and TSc
models are modified into sPT and sTEc models to estimate all allowance ratios
(AARs) of each DMU belonging to Ė, named DMU-o. DMU-o has a super-efficiency
larger than one and will deteriorate to 1 when DMU-o expands inputs and
deduces outputs. A DMU is superior to other efficient DMUs as it has a higher
super-efficiency. Variables $Q_{io}$ and $P_{ro}$ are changed to denote the
allowance ratios for DMU-o. In particular, DMU-o is excluded from the
reference set $J$ when assessing its super-efficiency.

###  3.1 Pure Technical Super-Efficiency (sPT) Model

Total adjustment price (TAP), the dual program of the sPT model:

$\delta_{o}^{sPT\star}=\min_{Q_{o},P_{o},\pi_{o}}\sum_{\forall i\in
I}Q_{io}\tau_{o}+\sum_{\forall r\in R}P_{ro}\tau_{o},\forall
o\in\textbf{\\.{E}};$ (54) $s.t.-\sum_{\forall j\in
J-\\{o\\}}x_{ij}\pi_{jo}+Q_{io}x_{io}\geq-x_{io},\forall i\in I;$ (55)
$\qquad\sum_{\forall j\in J-\\{o\\}}y_{rj}\pi_{jo}+P_{ro}y_{ro}=y_{ro},\forall
r\in R;$ (56) $\qquad\pi_{o},Q_{o},P_{o}\geq 0.$ (57)

Total virtual gap (TVG), the primal program of the sPT model:

$\Delta_{o}^{sPT\star}=\max_{v_{o},u_{o}}-\sum_{\forall i\in
I}v_{io}x_{io}+\sum_{\forall r\in R}u_{ro}y_{ro},\forall o\in\textbf{\\.{E}};$
(58) $\displaystyle s.t.-\sum_{\forall i\in I}v_{io}x_{ij}$
$\displaystyle+\sum_{\forall r\in R}u_{ro}y_{rj}\leq 0,\forall j\in
J-\\{o\\};$ (59) $x_{io}v_{io}\geq\tau_{o},\forall i\in I;$ (60)
$y_{ro}u_{ro}\geq\tau_{o},\forall r\in R;$ (61) $v_{o}\geq 0\quad and\quad
u_{o}\quad free.$ (62)

(59) limits each DMU-j, except DMU-o, upper bound to zero virtual gaps. While
(60) and (61) restrict virtual prices of DMU-o lower bound to the unified goal
price $\tau_{o}$.

Use Step I and Step II of the sPT model to determine the goal price, as shown
in (63). The estimated $u_{o}^{\\#}y_{o}$ equals $\$$1 and larger than
$v_{o}^{\\#}x_{o}$.

$\bar{t}=\$1/u_{o}^{\\#}y_{o}\textrm{ and }\tau_{o}^{\star}=\$\bar{t}.$ (63)

Similar to (11) and (12), the optimal solutions of (54) and (58) as depicted
in (64) and (65). The total virtual gaps (TVG) are equal to
$(-\alpha_{o}^{\\#}+\beta_{o}^{\\#})$ and
$(-\alpha_{o}^{\star}+\beta_{o}^{\star})$ in Steps I and-II .

$\displaystyle\$0<\delta_{o}^{sPT\\#}=\sum_{\forall i\in
I}Q_{io}^{\\#}\times\$1+\sum_{\forall r\in
R}P_{ro}^{\\#}\times\$1=\Delta_{o}^{sPT\\#};$ (64)
$\displaystyle\$0<\delta_{o}^{sPT\star}=\sum_{\forall i\in
I}Q_{io}^{\star}\tau_{o}^{\star}+\sum_{\forall r\in
R}P_{ro}^{\star}\tau_{o}^{\star}=\delta_{xo}^{sPT\star}+\delta_{yo}^{sPT\star}=\Delta_{o}^{sPT\star}<\$1.$
$\displaystyle\$0<\Delta_{j}^{sPT\\#}=-v_{o}^{\\#}x_{j}+u_{o}^{\\#}y_{j}=-\alpha_{j}^{\\#}+\beta_{j}^{\\#};$
(65)
$\displaystyle\$0<\Delta_{j}^{sPT\star}=-v_{o}^{\star}x_{j}+u_{o}^{\star}y_{j}=-\alpha_{j}^{\star}+\beta_{j}^{\star};j\in
J-\\{o\\}.$

######  Theorem 3.1.

The sPT model estimated sPT efficiency no less than 1.

Proof. The sPT model estimates the technical parameters $v_{o}^{\star}$ and
$u_{o}^{\star}$ to have the maximum virtual gap $\Delta_{o}^{sPT\star}$. Which
is converted into the super-pure-technical-efficiency (sPT) $E_{o}^{sPT\star}$
according to (LABEL:Eq:66).

$\displaystyle 1\leq E_{o}^{sPT\star}=\beta_{o}^{\star}/\alpha_{o}^{\star}.$
(66)

$E_{o}^{sPT\star}$ is no less than one, where $\alpha_{o}^{\star}\leq$
$\beta_{o}^{\star}=\$1$. $\square$
(67) likely represents this normalization process, where the solutions
obtained in Step I are adjusted or transformed to achieve the solutions in
Step II.

$\displaystyle(\delta_{o}^{sPT\star},\Delta_{o}^{sPT\star},v_{o}^{\star},u_{o}^{\star})=\bar{t}\times(\delta_{o}^{sPT\\#},\Delta_{o}^{sPT\\#},v_{o}^{\\#},u_{o}^{\\#}).$
(67)

The symbol $\mathcal{E}_{o}^{sPT}$ denotes the reference set of DMU-o in the
sPT evaluations. Each peer DMU-j belonging to the reference set has sPT
efficiency equals 1, and its estimated intensity, $\pi_{jo}^{\star}>0$. At the
same time, the estimated intensity of the other inefficient DMUs,
$\pi_{jo}^{\star}=0$. DMU-o estimates the benchmark of each input-i and
output-r, $\widehat{x}_{io}^{sPT\star}$ and $\widehat{y}_{ro}^{sPT\star}$ via
(68). DMU-o mimics the reference DMUs with their estimated intensities,
$\pi_{jo}^{\star}$, $\forall j\in\mathcal{E}_{o}^{sPT}$. DMU-o is superior to
its reference peers, whose criteria configurations are analog to DMU-o.

$\displaystyle\widehat{x}_{io}^{sPT\star}=\sum_{\forall
j\in\mathcal{E}_{o}^{sPT}}x_{ij}\pi_{jo}^{\star}=x_{io}(1+Q_{io}^{\star}),\forall
i\in I;$ (68) $\displaystyle\widehat{y}_{ro}^{sPT\star}=\sum_{\forall
j\in\mathcal{E}_{o}^{sPT}}y_{rj}\pi_{jo}^{\star}=y_{ro}(1-P_{ro}^{\star})\forall
r\in R.$

Assessing DMU-o with the adjusted input-i and output-r,
$\widehat{x}_{io}^{sPT\star}$ and $\widehat{y}_{ro}^{sPT\star}$, as shown in
(68), will have the sPT efficiency $\widehat{E}_{o}^{sPT\star}$ equals 1.

$\sum_{\forall j\in\mathcal{E}_{o}^{sPT}}\pi_{jo}^{\star}=\kappa_{o}^{1}.$
(69)

Let the total of estimated intensities be the first SIC scalar,
$\kappa_{o}^{1}$, computed via (69).

###  3.2 Super-Efficiency sTSc Model

Adding the SIC (73) to the sPT model’s dual program will have the next
program. The SIC corresponds to the free-in-sign decision variable,
$w_{o}^{c}$. TAP (dual) program of the sTSc model:

$\delta_{o}^{sTSc\star}=\min_{Q_{o},P_{o}}\sum_{\forall i\in
I}Q_{io}\tau_{o}+\sum_{\forall r\in R}P_{ro}\tau_{o},\forall
o\in\textbf{\\.{E}};$ (70) $s.t.-\sum_{\forall j\in
J-\\{o\\}}x_{ij}\pi_{jo}+Q_{io}x_{io}\geq-x_{io},\forall i\in I;$ (71)
$\sum_{\forall j\in J-\\{o\\}}y_{rj}\pi_{jo}+P_{ro}y_{ro}=y_{ro},\forall r\in
R;$ (72) $\sum_{\forall j\in J-\\{o\\}}\pi_{jo}=\kappa_{o}^{c};$ (73)
$\pi_{o},Q_{o},P_{o}\geq 0.$ (74)

TVG (primal) program of the sTSc model:

$\displaystyle\Delta_{o}^{sTSc\star}=\max_{x_{o},v_{o},w_{o}^{c}}-\sum_{\forall
i\in I}v_{io}x_{io}$ $\displaystyle+\sum_{\forall r\in
R}u_{ro}y_{ro}+\kappa_{o}^{c}w_{o}^{c},\forall o\in\textbf{\\.{E}};$ (75)
$\displaystyle s.t.-\sum_{\forall i\in I}v_{io}x_{ij}+\sum_{\forall r\in
R}u_{ro}y_{rj}+1w_{o}^{c}\leq 0,\forall j\in J-\\{o\\};$ (76)
$x_{io}v_{io}\geq\tau_{o},\forall i\in I;$ (77)
$y_{ro}u_{ro}\geq\tau_{o},\forall r\in R;$ (78) $v_{o}\geq
0,u_{o},w_{o}^{c}\quad free.$ (79)

Similar to the TSc model, the sTSc model determines the goal price in two
steps. $\tau_{o}^{\\#}=\$1$ and $\tau_{o}^{\star}=\$\bar{t}$ will solve the
TAP program comprehensively with the solutions
$(Q_{o}^{\star},P_{o}^{\star},\pi^{\star})=(Q_{o}^{\\#},P_{o}^{\\#},\pi^{\\#})$.
Use (27) to obtain $\gamma_{o}$ and $(1-\gamma_{o})$. Partition the vScalar
into two parts to reflect the effects of the SIC on inputs and outputs of
DMU-o.

The maximized TVG of DMU-o in Step II of (75) is expressed as the affected
virtual Output (avOutput) minus affected virtual Input (avInput), as shown in
(LABEL:Eq:80). The solutions in Step I have a similar expression.

$\displaystyle\$0<\Delta_{o}^{sTSc\star}=-v_{o}^{\star}x_{o}+u_{o}^{\star}y_{o}+\omega_{o}^{c\star}$
(80)
$\displaystyle=-[v_{o}^{\star}x_{o}-(1-\gamma_{o})\omega_{o}^{c\star}]+(u_{o}^{\star}y_{o}+\gamma_{o}\omega_{o}^{c\star})$
$\displaystyle=-[gvInput-ivScalar]+[gvOutput+ovScalar]$
$\displaystyle=-avInput+avOutput=-\alpha_{o}^{c\star}+\beta_{o}^{c\star}<\$1;$
$\displaystyle\$0<\Delta_{o}^{sTSc\\#}=-\alpha_{o}^{c\\#}+\beta_{o}^{c\\#}.$

Similarly, the estimated vScalar of Step II in (76), $1w_{o}^{*}$, is
decomposed into two components, as shown in (LABEL:Eq:81); the solutions of
Step I have the similar expressions.

$\displaystyle\$0<\Delta_{j}^{sTSc\star}=-v_{o}^{\star}x_{j}+u_{o}^{\star}y_{j}+1w_{o}^{c\star}$
(81)
$\displaystyle=-[v_{o}^{\star}x_{j}-(1-\gamma_{o})1w_{o}^{c\star}]+(u_{o}^{\star}y_{j}+\gamma_{o}1w_{o}^{c\star})=-\alpha_{j}^{c\star}+\beta_{j}^{c\star},\forall
j\in J,j\neq{o}.$

Each DMU-j is expressed by the pair of _(avInput, avOutput)_ , the virtual
scales. The symbol $\mathcal{E}_{o}^{sTS}$ denotes the set of reference peers
in evaluating DMU-o by the sTS model, where each reference peer DMU-j has
$\pi_{jo}^{\star}>0$. The other inefficient DMU-j has $\pi_{jo}^{\star}=0$. In
(LABEL:Eq:81), any best DMU-j belongs to $\mathcal{E}_{o}^{sTS}$ has
$\alpha_{j}^{c\\#}=\beta_{j}^{c\\#}$ and
$\alpha_{j}^{c\star}=\beta_{j}^{c\star}$, while the other remaining DMUs have
$\alpha_{j}^{c\\#}>\beta_{j}^{c\\#}$ and
$\alpha_{j}^{c\star}>\beta_{j}^{c\star}$.

######  Theorem 3.2.

The sTSc model estimates sTSc efficiency score larger than 1.

Proof. Using (82) would have the dimensionless value of $\bar{t}$.

$\displaystyle\$\bar{t}=\$1/\beta_{o}^{c\\#}=\$1/(u_{o}^{\\#}y_{o}+\gamma_{o}\omega_{o}^{c\\#})=\tau_{o}^{\star}.$
(82)

Because $\tau_{o}^{\\#}:\tau_{o}^{\star}=\$1:\$\bar{t}$, therefore, if
$\tau_{o}^{\\#}$ = $\$\bar{t}$, then $\beta_{o}^{c\star}$ equals $1. According
to (LABEL:Eq:80), the avInput and avOutput, $\alpha_{o}^{c\star}$ and
$\beta_{o}^{c\star}$, contain the gvInput, gvOutput, ivScalar, and ovScalar.
$-\alpha_{o}^{c\star}$ \+ $\beta_{o}^{c\star}$ is larger than $0. Using (83),
the estimated virtual gap $\Delta_{o}^{sTSc\star}$ is converted into the
maximized sTSc super-efficiency score, $E_{o}^{sTSc\star}$, should be larger
than 1.

$\displaystyle
E_{o}^{sTSc\star}=\frac{u_{o}^{\star}y_{o}+\gamma_{o}\omega_{o}^{c\star}}{v_{o}^{\star}x_{o}-(1-\gamma_{o})\omega_{o}^{c\star}}=\frac{\beta_{o}^{c\star}}{\alpha_{o}^{c\star}}>1.$
(83)

$\square$

Normalizing the solutions of Step I by the dimensionless value $\bar{t}$ would
obtain the solutions of Step II, as shown in (84).

$\displaystyle(\Delta_{o}^{sTSc\star},\delta_{o}^{sTSc\star},v_{o}^{\star},u_{o}^{\star},w_{o}^{c\star})=\quad\bar{t}\times(\Delta_{o}^{sTSc\\#},\delta_{o}^{sTSc\\#},v_{o}^{\\#},u_{o}^{\\#},w_{o}^{c\\#}).$
(84)

Use (85) to compute the benchmark of each performance index,
$\widehat{x}_{io}^{sTSc\star}$ , and $\widehat{y}_{ro}^{sTSc\star}$. DMU-o
imitates the reference peers with their estimated intensities
$(\pi_{jo}^{\star})$.

$\displaystyle\widehat{x}_{io}^{sTSc\star}$ $\displaystyle=\sum_{\forall
j\in\mathcal{E}_{o}^{sTS}}x_{ij}\pi_{jo}^{\star}=x_{io}(1+Q_{io}^{\star}),\forall
i\in I;$ (85) $\displaystyle\widehat{y}_{ro}^{sTSc\star}$
$\displaystyle=\sum_{\forall
j\in\mathcal{E}_{o}^{sTS}}y_{rj}\pi_{jo}^{\star}=y_{ro}(1-P_{ro}^{\star}),\forall
r\in R.$

######  Theorem 3.3.

DMU-o has an sTSc super-efficiency equals 1 when using the estimated
benchmarks.

Proof. DMU-o uses the benchmarks, $\widehat{x}_{io}^{TSc\star}$ and
$\widehat{y}_{ro}^{TSc\star}$, will decrease the sTSc super-efficiency
$\widehat{E}_{o}^{sTSc\star}$ to 1 while the $\delta_{o}^{TSc\star}$ drops to
$\$$0\. $\square$

###  3.3 The SIC Effects in Measuring the Super-Efficiency

In Figure 4, the coordinates of points O, AP1, AP2, B1, and B2 of the sTS1 and
sTS2 models in evaluating DMU-B can be visualized. (86),(87), and (88) compute
the slopes of the vectors $\overrightarrow{\rm O,Bc}$, $\overrightarrow{\rm
O,APc}$, and $\overrightarrow{\rm APc,Bc}$, respectively, where the choice ”c”
could be ”1” and ”2.”

$\displaystyle\bar{m}(\overrightarrow{\rm
O,Bc})=\frac{{\beta_{o}^{c\star}-0}}{{\alpha_{o}^{c\star}-0}}=\frac{{\beta_{o}^{c\star}}}{{\alpha_{o}^{c\star}}}.$
(86) $\displaystyle\bar{m}(\overrightarrow{\rm
O,APc})=\frac{\gamma_{o}\omega_{o}^{c\star}-0}{-(1-\gamma_{o})\omega_{o}^{c\star}-0}=\frac{\gamma_{o}\omega_{o}^{c\star}}{-(1-\gamma_{o})\omega_{o}^{c\star}}.$
(87) $\displaystyle\bar{m}(\overrightarrow{\rm
APc,Bc})=\frac{\beta_{o}^{c\star}-\gamma_{o}\omega_{o}^{c\star}}{{\alpha_{o}^{c\star}}-[-(1-\gamma_{o})\omega_{o}^{c\star}]}=\frac{u_{o}^{\star}y_{o}}{v_{o}^{\star}x_{o}}.$
(88)

The same analysis should apply to DMU-D, as Figure 4 depicts.

######  Theorem 3.4.

The sTSc model fulfills Condition C3 that the relative efficiency are affected
by the SIC.

Proof. The three vectors have the relationships as shown in (89). The
estimated sTSc super-efficiency score is affected by the SIC.

$\displaystyle\overrightarrow{\rm O,Bc}=\overrightarrow{\rm
O,APc}+\overrightarrow{\rm APc,Bc};\quad
E_{o}^{sTSc\star}=\bar{m}(\overrightarrow{\rm
O,Bc})=\frac{\beta_{o}^{c\star}}{\alpha_{o}^{c\star}}.$ (89)

The effects of the SIC within the sTSc model could be visualized as triangular
with points O, APc, and Bc in the 2D graphical intuition. The slopes
$\bar{m}(\overrightarrow{\rm O,APc})$ and $\bar{m}(\overrightarrow{\rm
APc,Bc})$ are not the scale and technical efficiencies. $\square$

##  4 Four-phase VGA Procedure

The following presentation is to calibrate the AARs to measure the
inefficiency of DMU-o. It can be applied to measure super-efficiency.
Nonetheless, the PT model estimated AARs may not be achievable for DMU-o. The
TSc model with the SIC equates to the choice scalar, $\kappa_{o}^{c},$ that
identifies the reference peers in $\mathcal{E}_{o}^{TS}$. The benchmarks of
inputs and outputs are the function of intensities of the reference peers
belonging to $\mathcal{E}_{o}^{TS}$, as shown in (33). Section 2.5.4
illustrates the effects of SIC on the solutions.

The four-phase process illustrated in Figure 1 significantly improves the
productivity management of DMUs with Conditions C1, C2, C3, and C4.

In summary, the four conditions collectively guide the estimation of AARs by
recognizing efficiency’s inherent and operational dimensions, promoting
continuous improvement, and setting realistic targets for DMU-o.

This framework is structured around four conditions that epitomize a
performance improvement problem. It incorporates practical considerations into
the PT and TSc VGA models for systematic solutions. By integrating innovations
that cater to the third and fourth conditions, our approach offers a more
comprehensive and systematic method for efficiency assessment with potential
applicability across various sectors.

###  4.1 Phase 1: PT model Identifies the First Scalar.

An initial assessment distinguishes between efficient and inefficient DMUs in
the performance evaluation. This distinction is crucial for the subsequent
analysis and is primarily based on the existence of a ”virtual gap.” An
inefficient DMU-o is characterized by a positive virtual gap, indicating a
discrepancy between its current performance and the Efficiency Equator. The
DMU must undergo a comprehensive four-phase improvement process.

Figure 1: Four-phase VGA procedure.

The commencement of this process involves the identification of the first SIC
scalar, $\kappa_{o}^{1}$. This scalar is conceptualized as the sum of the
estimated intensities associated with DMU-o’s efficient counterparts,
$\kappa_{o}^{1}=\sum_{\forall j\in\mathcal{E}_{o}^{PT}}\pi_{jo}^{\star}$.
Essentially, it serves as a quantitative measure of the extent to which DMU-o
needs to adjust its operations to align with the practices of its reference
peers.

Conversely, a DMU identified as efficient exhibits a zero virtual gap,
signifying that its performance already aligns with the best practices
observed across the dataset at the optimal level. Such DMUs are deemed
ineligible for further analysis within this framework. For these units, the
second scenario presented in Section 3 should be employed for continuous
performance management and improvement, ensuring they maintain their
efficiency status over time.

###  4.2 Phase 2: TS1 Model Uses the First Scalar.

The comparison between the optimal solutions of the PT and TS1 models in Steps
I and II reveals some critical insights. Both models share identical TAP
solutions in Step I due to the equality $\tau_{o}^{PT\\#}=\tau_{o}^{TS1\\#}$
and $\sum_{\forall j\in\mathcal{E}_{o}^{PT}}\pi_{jo}^{\\#}$= $\sum_{\forall
j\in\mathcal{E}_{o}^{TS1}}\pi_{jo}^{\\#}$ = $\kappa_{o}^{1}$. However, their
TVG programs yield two sets of optimal solutions due to the additional
vScalar. This process leads to a specific relationship between their
intensities, resulting in disparate benchmarks for performance indices between
the two models.

######  Theorem 4.1.

Step I of PT and TS1 models are linked.

Proof. (90) shows Step I of the PT and TS1 models have the relationships.

$\displaystyle\Delta_{o}^{TS1\\#}=\Delta_{o}^{PT\\#}=\delta_{o}^{TS1\\#}=\delta_{o}^{PT\\#};(Q_{o}^{PT\\#},P_{o}^{PT\\#})=(Q_{o}^{TS1\\#},P_{o}^{TS1\\#});$
(90)
$\displaystyle(v_{o}^{PT\\#},u_{o}^{PT\\#})\neq(v_{o}^{TS1\\#},u_{o}^{TS1\\#});\sum_{\forall
j\in\mathcal{E}_{o}^{PT}}\pi_{jo}^{\\#}=\sum_{\forall
j\in\mathcal{E}_{o}^{TS1}}\pi_{jo}^{\\#}=\kappa_{o}^{1}.$

Note the best peers’ intensities, $\pi_{jo}^{PT\\#}$, and $\pi_{jo}^{TS1\\#}$,
may be different; therefore, the benchmarks of performance indices of the two
models are distant. They have identical solutions on the dual variables. But
have different primal solutions. $\square$

Step II further differentiates the two models. Although their TAP programs
provide the same total values, their goal prices and other variables differ.
This disparity leads to different efficiency measurements: $E_{o}^{PT\star}$
represents pure technical efficiency, and $E_{o}^{TS1\star}$ comprises the
effects on the gray virtual input and output and scalar prices. Notably, the
SIC scalar proves to be a crucial factor in evaluating DMU-o. In Step II of PT
and TS1 models, the TAP programs equal to ($\sum_{\forall i\in I}$
$Q_{io}^{\star}$ \+ $\sum_{\forall r\in R}$ $P_{ro}^{\star}$ ) multiplies the
distinct virtual goal prices, $\tau_{o}^{PT\star}$ and $\tau_{o}^{TS1\star}$.
The PT and TS1 models have the relationships shown in (91).

$\displaystyle\tau_{o}^{TS1\star}\neq\tau_{o}^{PT\star},\Delta_{o}^{TS1\star}\neq\Delta_{o}^{PT\star};\quad(v_{o}^{PT\star},u_{o}^{PT\star})\neq(v_{o}^{TS1\star},u_{o}^{TS1});$
(91)
$\displaystyle(Q_{o}^{PT\star},P_{o}^{PT\star})=(Q_{o}^{TS1\star},P_{o}^{TS1\star});\quad
0<E_{o}^{PT\star}\neq E_{o}^{TS1\star}\leq 1.$

###  4.3 Phase 3: Identifying the Second Scalar.

The standard sensitivity analysis of the LP TS1 model involves perturbing the
$\kappa_{o}^{1}$ scalar, which results in allowable decreases and increases.
Specifically, $\kappa_{o}^{2}$ equals $\kappa_{o}^{1}$ minus the allowable
decrease or $\kappa_{o}^{2}$ equals $\kappa_{o}^{1}$ plus the allowable
increase, as expressed in (92).

$\displaystyle\kappa_{o}^{2}$ $\displaystyle=\kappa_{o}^{1}-\textrm{(allowable
decreasing of }\kappa_{o}^{1}),$ (92) $\displaystyle\textrm{or
}\kappa_{o}^{2}$ $\displaystyle=\kappa_{o}^{1}+\textrm{(allowable increasing
of }\kappa_{o}^{1}).$

######  Theorem 4.2.

The final SIC scalar $\kappa_{o}^{c}$ in Phase-4 is linked to $\kappa_{o}^{1}$
in Phase-1.

Proof. These bounds ($\kappa_{o}^{1}$ and $\kappa_{o}^{2}$) represent the
possible range of SIC scalars ($\kappa_{o}^{c}$) within the TSc model. If
$w_{o}^{TS1\star}$ and $w_{o}^{TS2\star}$ are greater than 0,
$\kappa_{o}^{1}<\kappa_{o}^{c}<\kappa_{o}^{2}$. Otherwise,
$\kappa_{o}^{1}>\kappa_{o}^{c}>\kappa_{o}^{2}$. $\square$

The best peers in the TS1 and TS2 models remain consistent, as indicated by
$\mathcal{E}_{o}^{TS1}$=$\mathcal{E}_{o}^{TS2}$=$\mathcal{E}_{o}^{TSc}$.
However, the estimated intensities of these peers differ, $\pi^{TS1\star}$
$\neq\pi^{TS2\star}$. The solutions of these models exhibit relationships
detailed in (93), displaying discrepancies in the estimated variables, such
as:

$\displaystyle\tau_{o}^{TS1\star}\neq\tau_{o}^{TS2\star};\quad(v_{o}^{TS1\star},u_{o}^{TS1\star})\neq(v_{o}^{TS2\star},u_{o}^{TS2\star});$
(93) $\displaystyle w_{o}^{TS1\star}\neq
w_{o}^{TS2\star};\quad(Q_{o}^{TS1\star},P_{o}^{TS1\star})\neq(Q_{o}^{TS2\star},P_{o}^{TS2\star}).$

###  4.4 Phase 4: Select the Final Scalar Within the Region.

Phases 1, 2, and 3 integration confirms DMU-o’s learning process with the
reference peers, whose total intensities are bounded within the
$\kappa_{o}^{1}$ and $\kappa_{o}^{2}$. Phase 4, a critical decision-making
step, involves selecting the final scalar $\kappa_{o}^{z}$ for the TSz model.
DMU-o undertakes several trials in choosing scalars in Phase 4 to determine a
preferred final scalar, $\kappa_{o}^{z}$. Each choice scalar corresponds to a
unique set of estimated intensities and benchmarks of inputs and outputs,
$\widehat{x}_{io}^{TSz}$ and $\widehat{y}_{ro}^{TSz}$ derived from the
efficient peers.

By emulating the reference peers based on the estimated intensities, DMU-o
gains insights into feasible and desirable productivity benchmarks. This
interactive approach ensures a more thorough evaluation of DMU-o. Exploring
different SIC scalars and model variations aids in comprehending the diverse
impacts these factors have on assessments. Ultimately, this process helps make
better-informed management decisions regarding productivity and efficiency.

######  Theorem 4.3.

DMU-o chooses the final SIC scalar $\kappa_{o}^{z}$ to fulfill Condition C4,
the estimated AARs are applicable.

Proof. DMU-o may redefine the datasets by altering the performance indices and
DMUs and repeating the four-phase procedure. $\square$

##  5 Numerical examples and the 2D geometric intuitions

DMU-o (DMU-K) evaluates its performance relative to compatible peers such as
DMUs K, A, B, D, G, and H. It selects inputs x1 and x2 and outputs y1 and y2,
measured in various units. Some market prices per unit of the inputs and
outputs are not attainable. This assessment disregards the interactions
between the four indices. DMU-o supplies the dataset detailed in Tables 1,
which other DMUs may accept and use for their evaluations.

Table 1: The example data. DMU-j | K | A | B | D | G | H
---|---|---|---|---|---|---
$x_{1j}$(ton) | 1.6 | 2.3 | 1 | 1.9 | 1.8 | 2.5
$x_{2j}$(hr) | 145 | 120 | 29 | 281 | 250 | 100
$y_{1j}$$(m^{3})$ | 1036 | 1327 | 567 | 2446 | 1794 | 1000
$y_{2j}$($\%$) | 49 | 97 | 89 | 97 | 57 | 70

Table 2 summarizes the inefficiency measurements of DMU-o within the first
scenario by the PT, TS1, TS2, and TS3 models. Model PT evaluates DMU-B and
DMU-D and confirms they are efficient units. Section 5.2 offers the second
scenario to measure the super-efficiencies of efficient DMUs.

###  5.1 Measures the Inefficiencies

####  5.1.1 Relationships Between PT and TS1 Models in Step I

(18) and (35) are used to calculate the total adjustment prices
($\delta_{o}^{PT\\#}$ = $\delta_{o}^{TS1\\#}$) and total virtual gaps
($\Delta_{o}^{PT\\#}$= $\Delta_{o}^{TS1\\#}$) using specific data provided in
row R2 of Table 2. Each amount is $2.3010. Additionally, the TS1-I model
incorporates the SIC impact through the vScalar, $\omega_{o}^{1\\#}$(=$2.479),
in calculating the total virtual gap. This result demonstrates how the virtual
gap, considering the SIC and vScalar in the TS1-I model, maintains the same
value as in the PT-I model, indicating a consistent evaluation despite
including additional factors.

Figure 2: Evaluate DMU-K by PT and TS3 models.

####  5.1.2 Compare the PT and TS3 Models

By try and error, we identified $\kappa_{o}^{3}$ equals 0.718 to have
$E_{o}^{PT\star}$=$E_{o}^{TS3\star}$(=0.589). The columns of PT-II and TS3-II
in Table 2 list the estimated solutions, in which rows R8 and R9 are the two
virtual technology sets, ($\alpha_{j}^{\star},\beta_{j}^{\star}$) and
($\alpha_{j}^{3\star},\beta_{j}^{3\star}$), where DMU-j= (A, B, D, G, H, K),
as per (12), (28), and (29).

Via (43) and (44), the locations of Tp,
($\widehat{\alpha}_{o}^{\star},\widehat{\beta}_{o}^{\star}$) and point T3,
($\widehat{\alpha}_{o}^{3\star},\widehat{\beta}_{o}^{3\star}$) on the
Efficiency Equator are obtained. The estimated rectilinear distances from
points Kp to Tp and from point K3 to T3 are $\delta_{o}^{PT\star}$, and
$\delta_{o}^{TS3\star}$, see the expressions in the upper-left corner of
Figure 2. Section 2.5.4 illustrated the SIC affected the estimations as shown
in (52). In Figure 2, each point signifies a DMU-j within the virtual
technology sets of the PT-II ($\Phi_{o}^{PT\star}$) and TS3-II models
($\Phi_{o}^{TS3\star}$), respectively. The points (Ap, Bp, Dp, Gp, Hp, Kp) and
(A3, B3, D3, G3, H3, K3) located at ($\alpha_{j}^{\star},\beta_{j}^{\star}$)
and ($\alpha_{j}^{3\star},\beta_{j}^{3\star}$). The blue dashed lines are
graphically representing the vectors $\overrightarrow{\rm O,K3}$,
$\overrightarrow{\rm O,AP3}$, and $\overrightarrow{\rm AP3,K3}$. The red
dashed line indicates the vector $\overrightarrow{\rm O,Kp}$ of pure technical
efficiency, $E_{o}^{PT\star}$= $\beta_{o}^{3\star}$/$\alpha_{o}^{3\star}$.

####  5.1.3 Compare the Solutions between TS1 and TS2 models

Figure 3, similar to Figure 2, depicts the locations of DMUs in the TS1-II and
TS2-II models, respectively, (A1, B1, D1, G1, H1, K1) and (A2, B2, D2, G2, H2,
K2). Their coordinates, ($\alpha_{j}^{1\star},\beta_{j}^{1\star}$) and
($\alpha_{j}^{2\star},\beta_{j}^{2\star}$), are listed in rows R8 and R9. DMUs
B and D emerge as efficient peers with their intensities $\pi_{B}^{\star}$ and
$\pi_{D}^{\star}$. R4 detailed the benchmarks of inputs and outputs. R5 lists
the virtual prices. R6 contains the benchmark virtual scales. R7 lists
$\Xi_{o}^{TSc\star}$ and $E_{o}^{TSc\star}$. Since $w_{o}^{\star}>0$,
decreasing $\kappa_{o}^{1}$ (=1.5153) to $\kappa_{o}^{2}$ (=0.515), DMU-o
exhibits decreasing RTvS:
$\Xi_{o}^{TS1\star}(=2.435)>\Xi_{o}^{TS2\star}(=1.497)$. At the same time, the
relative efficiency is increasing, $E_{o}^{TS1\star}$ (=0.411) less than
$E_{o}^{TS2\star}$ (=0.688). However, DMU-o may use (48) to analyze the
interlinkage relationships between input and output indices for managing the
performance indices.

The TSc program introduces an anchor point (APc) located at the point
$[(1-\gamma_{o})\kappa_{o}^{c}w_{o}^{\star}$,
$-\gamma_{o}\kappa_{o}^{c}w_{o}^{\star}]$, denotes as [AP(x), AP(y)]. Anchor
points are located in the second and fourth quadrants when $w_{o}^{\star}>0$
and $w_{o}^{\star}<0$. The rectilinear distance between point APc and the
origin O is the vScalar, $\kappa_{o}^{c}w_{o}^{\star}$. In Figure 3, the
anchor points of the TS1 and TS2 models are located at AP1 and AP2. Because
$w_{o}^{\star}>0$, the two anchor points are located in the fourth quadrant.

The TS3 model interpolates TS1 and TS2 models, providing a framework for
determining the optimal scalars $\kappa_{o}^{3}$ within the interval of SIC
scalars. Using the scalar $\kappa_{o}^{3}$ between $\kappa_{o}^{1}$ and
$\kappa_{o}^{2}$, the TS3 model has the anchor points, and AP3 is located
between AP1 and AP2. Inefficient DMUs are located under the Efficiency
Equator. DMU-j lies in the fourth quadrant if, according to (2), one could
have $\Delta_{j}^{TSc\star}>\$0$.
($\alpha_{j}^{c\star}>\$0,\beta_{j}^{c\star}<\$0$), and their ratio is less
than zero. Conversely, DMU-j is in the third quadrant if
($\alpha_{j}^{c\star}<\$0,\beta_{j}^{c\star}<\$0$), and their ratio is greater
than zero.

Section 2.5.4 illustrated the SIC affected the estimations as shown in (53).
Therefore, eliminating an inefficient DMU-j from (X, Y) does not impact the
evaluation. Using geometric vectors, we can visually interpret the effects
under the SIC scalars $\kappa_{o}^{1}$ and $\kappa_{o}^{2}$. In Figure 3, the
purple color dashed lines express the three vectors $\overrightarrow{\rm
O,K1},\overrightarrow{\rm AP1,O}$, and $\overrightarrow{\rm AP1,K1}$ of the
TS1 model can be found.

Similarly, the green color dashed lines express the three vectors
$\overrightarrow{\rm O,K2},\overrightarrow{\rm AP2,O}$, and
$\overrightarrow{\rm AP2,K2}$ of the TS2 model. The three vectors
$\overrightarrow{\rm O,K3}$, $\overrightarrow{\rm AP3,O}$ and
$\overrightarrow{\rm AP3,K3}$ of TS3 model are not depicted for simplicity.

Table 2: Evaluate Inefficiencies for DMU-K, DMU-B, and DMU-D. | ’DMU-o=’ |  | K | K | K | K | K | K | K | B | D
---|---|---|---|---|---|---|---|---|---|---|---
Row | Solution | Unit | PT I | PT II | TS1 I | TS1 II | TS2 I | TS2 II | TS3 II | PT II | PT II
R1 | $\kappa_{o}^{c}$ | - | 1.5153 | 1.5153 | 1.5153 | 1.5153 | 0.5150 | 0.5150 | 0.718 | 1 | 1
| $\tau_{o}$ | $\$$ | 1 | 0.179 | 1.000 | 0.256 | 1.000 | 0.500 | 0.413 | 0.500 | 0.266
R2 | $\Delta_{o},\delta_{o}$ | $ | 2.3010 | 0.4113 | 2.3010 | 0.5893 | 0.6643 | 0.3321 | 0.411 | 0 | 0
| $v_{1o}$ | $/ton | 2.8713 | 0.5133 | 0.6250 | 0.1601 | 0.6250 | 0.3125 | 0.258 | 0.500 | 0.387
| $v_{2o}$ | $/hr | 0.0069 | 0.0012 | 0.0069 | 0.0018 | 0.0069 | 0.0034 | 0.003 | 0.017 | 0.001
| $u_{1o}$ |  $/$m^{3}$ | 0.0022 | 0.0004 | 0.0011 | 0.0003 | 0.0011 | 0.0006 | 0.0005 | 0.001 | 0.0003
| $u_{2o}$ |  $ / $\%$ | 0.0204 | 0.0036 | 0.0204 | 0.0052 | 0.0204 | 0.0102 | 0.008 | 0.006 | 0.003
| $w_{o}$ | $ | 0 | 0 | 1.6362 | 0.4190 | 1.6362 | 0.8181 | 0.675 | 0 | 0
R3 | $Q_{1o}$ | - | 0 | 0 | 0 | 0 | 0.4554 | 0.4554 | 0.363 | 0 | 0
| $Q_{2o}$ | - | 0.5334 | 0.5334 | 0.5334 | 0.5334 | 0.2089 | 0.2089 | 0.275 | 0 | 0
| $P_{1o}$ | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| $P_{2o}$ | - | 1.7677 | 1.7677 | 1.7677 | 1.7677 | 0 | 0 | 0.359 | 0 | 0
| $\pi_{B}$ | - | 1.421 | 1.421 | 1.421 | 1.421 | 0.119 | 0.119 | 0.383 | 1 | 0
| $\pi_{D}$ | - | 0.094 | 0.094 | 0.094 | 0.094 | 0.396 | 0.396 | 0.335 | 0 | 1
R4 | $\widehat{x}_{1o}$ | ton | 1.6 | 1.6 | 1.6 | 1.6 | 0.8713 | 0.8713 | 1.019 | 1.0 | 1.9
| $\widehat{x}_{2o}$ | hr | 67.66 | 67.66 | 67.66 | 67.66 | 114.72 | 114.72 | 105.165 | 29 | 281
| $\widehat{y}_{1o}$ | $m^{3}$ | 1036 | 1036 | 1036 | 1036 | 1036 | 1036 | 1036 | 567 | 2446
| $\widehat{y}_{2o}$ | % | 135.6 | 135.6 | 135.6 | 135.6 | 49.0 | 49.0 | 66.58 | 89 | 97
| $\gamma_{o}$ | - | 0.232 | 0.232 | 0.232 | 0.232 | 1.000 | 1.000 | 0.640 | 0 | 0
| $\omega_{o}^{a}$ | $ | 0 | 0 | 2.479 | 0.635 | 0.843 | 0.421 | 0.485 | 0 | 0
R5 | $x_{1o}v_{1o}$ | $ | 4.594 | 0.821 | 1.000 | 0.256 | 1.578 | 0.789 | 0.689 | 0.500 | 0.734
| $x_{2o}v_{2o}$ | $ | 1.000 | 0.179 | 3.479 | 0.891 | 1.265 | 0.632 | 0.622 | 0.500 | 0.266
| $y_{1o}u_{1o}$ | $ | 2.293 | 0.410 | 1.178 | 0.302 | 1.178 | 0.589 | 0.486 | 0.500 | 0.734
| $y_{1o}u_{1o}$ | $ | 1.000 | 0.179 | 3.479 | 0.891 | 1.843 | 0.921 | 0.898 | 0.500 | 0.266
R6 | $\alpha_{o},\alpha_{o}^{c}$ | $ | 5.594 | 1.000 | 3.905 | 1.000 | 2.000 | 1.000 | 1.000 | 1 | 1
| $\beta_{o},\beta_{o}^{c}$ | $ | 3.293 | 0.589 | 1.604 | 0.411 | 1.336 | 0.668 | 0.589 | 1 | 1
| $\widehat{\alpha}_{o},\widehat{\alpha}_{o}^{c}$ | $ | 5.061 | 0.905 | 3.371 | 0.863 | 1.336 | 0.668 | 0.737 | 1 | 1
| $\widehat{\beta}_{o},\widehat{\beta}_{o}^{c}$ | $ | 5.061 | 0.905 | 3.371 | 0.863 | 1.336 | 0.668 | 0.737 | 1 | 1
R7 | $\Xi_{o}$ | - | 1.699 | 1.699 | 2.435 | 2.435 | 1.497 | 1.497 | 1.699 | 1 | 1
| $E_{o}$ | - | 0.589 | 0.589 | 0.411 | 0.411 | 0.668 | 0.668 | 0.589 | 1 | 1
R8 | $\alpha_{K},\alpha_{K}^{c}$ | $ | 5.594 | 1 | 3.905 | 1.000 | 2 | 1 | 1.000 | 3.300 | 0.755
| $\alpha_{A},\alpha_{A}^{c}$ | $ | 7.432 | 1.000 | 3.522 | 0.902 | 2.265 | 1.133 | 0.902 | 3.219 | 1.002
| $\alpha_{B},\alpha_{B}^{c}$ | $ | 3.071 | 1.328 | 2.082 | 0.533 | 0.825 | 0.413 | 0.533 | 1 | 0.414
| $\alpha_{D},\alpha_{D}^{c}$ | $ | 7.393 | 0.549 | 4.382 | 1.122 | 3.125 | 1.563 | 1.122 | 5.795 | 1
| $\alpha_{G},\alpha_{G}^{c}$ | $ | 6.892 | 1.322 | 4.106 | 1.052 | 2.849 | 1.425 | 1.052 | 5.210 | 0.932
| $\alpha_{H},\alpha_{H}^{c}$ | $ | 7.868 | 1.232 | 3.509 | 0.899 | 2.252 | 1.126 | 0.899 | 2.974 | 1.061
R9 | $\beta_{K},\beta_{K}^{c}$ | $ | 3.293 | 0.589 | 1.604 | 0.411 | 1.336 | 0.668 | 0.589 | 1.189 | 0.445
| $\beta_{A},\beta_{A}^{c}$ | $ | 4.917 | 0.879 | 3.110 | 0.796 | 1.853 | 0.926 | 0.796 | 1.715 | 0.664
| $\beta_{B},\beta_{B}^{c}$ | $ | 3.071 | 0.549 | 2.082 | 0.533 | 0.825 | 0.413 | 0.533 | 1 | 0.414
| $\beta_{D},\beta_{D}^{c}$ | $ | 7.393 | 1.322 | 4.382 | 1.122 | 3.125 | 1.563 | 1.122 | 2.702 | 1
| $\beta_{G},\beta_{G}^{c}$ | $ | 5.134 | 0.918 | 2.824 | 0.723 | 1.568 | 0.784 | 0.723 | 1.902 | 0.695
| $\beta_{H},\beta_{H}^{c}$ | $ | 3.642 | 0.651 | 2.187 | 0.560 | 0.930 | 0.465 | 0.560 | 1.275 | 0.492
R10 | AP(x) | $ | - | - | 1.905 | 0.488 | 0 | 0 | 0.175 | - | -
| AP(y) | $ | - | - | -0.575 | -0.147 | -0.843 | -0.421 | -0.310 | - | -

Point K3 is located between points K1 and K2 on the normalization vertical
line (N, E). Using the final SIC scalar $\kappa_{o}^{z}$ in the TSz model,
points APz, Kz, and Tz are located between points (AP1 and AP2), (K1 and K2),
and (T1 and T2), respectively. The three vectors, $\overrightarrow{\rm O,Kz}$,
$\overrightarrow{\rm APz,O}$, $\overrightarrow{\rm APz,Kz}$, shall be
visualized in Figure 3, similar to (53). The upper left corner of the figure
depicts the rectilinear distances of DMU-o to project on the Efficiency
Equator.

Figure 3: Evaluate DMU-K by TS1 and TS2 models.

###  5.2 Measures the Super-efficiencies

The first scenario confirmed DMU-B and DMU-D are efficient. The second
scenario employs the sPT and sTSc models to measure their super-efficiencies
in the four-phase procedure. Table LABEL:table:3 summarized the solutions of
the sPT, sTS1, sTS2, and sTSz models. To illustrate the fourth phase, we
assume the final SIC scalars are the mid-point of the interval of
$\kappa_{o}^{1}$ and $\kappa_{o}^{2}$. The anchor point of the sTSz model that
contains the SIC scalar $\kappa_{o}^{z}$, APz is located in the region of
points O, AP1, and AP2 because DMU-o is excluded from (59). In Figure 4, AP1
and AP2 are in the second quadrant because $\omega_{o}^{c}>\$0$. When
$\omega_{o}^{c}>\$0$, the anchor point will be located in the fourth quadrant.

R6 in Table LABEL:table:3 listed the locations of DMU-o and its projection
points on the Efficiency Equator. DMU-o is located in the first quadrant and
above the Efficiency Equator. DMU-o projects on the Efficiency Equator, the T1
and T2 in Figure 4 and 5. The reference DMUs of DMU-B, A1, and A2 are on the
Efficiency Equator, while the remaining inefficient DMUs lie under it. In
evaluating DMU-D, DMU-B, and DMU-G, they are located on the Efficiency
Equator.

Figure 4: Evaluate DMU-B by sTS1 and sTS2 Models.

In Table LABEL:table:3, DMU-B has higher super-efficiencies (see R7), which
reduced the outputs will have an efficiency score equal to 1. Regarding EA,
DMU-B has strong outputs and can be selected as the best alternative to the
MCDM problem.

##  6 Discussions

Applying the four-phase procedure to an MCDM problem, the decision-maker needs
to manipulate the SIC scalar $\kappa_{o}^{z}$ for each DMU-o. The decision-
maker performs the sensitivity analysis to compare the choices. The efficient
DMUs may not be selected since the uncertainties could not always be expressed
in the data matrix and the EA solutions. Our contributions to the MCDM
problems are as follows. [(ii)]

Integration for Enhanced Decision Support: MCDM can incorporate efficiency
analysis results as criteria, providing a quantitative basis for evaluating
alternatives. For instance, VGA results can serve as efficiency scores within
an MCDM framework to rank and select the most efficient DMUs or alternatives.

Complementary in Performance Improvement: While MCDM facilitates the selection
among alternatives based on multiple criteria, efficiency analysis identifies
gaps and potential areas for deterioration. Integrating these approaches can
guide strategic decisions on resource allocation, process optimization, and
performance enhancement.

Figure 5: Evaluate DMU-D by sTS1 and sTS2 Models.

Framework for Complex Decisions: In complex decision scenarios, such as
strategic planning, resource management, and system design, combining MCDM
with efficiency analysis offers a robust framework for considering a broad
spectrum of performance metrics and stakeholder preferences, thereby
supporting more informed and comprehensive decisions.

Methodological Cross-Fertilization: Advances in MCDM and efficiency analysis
methodologies can enrich each other. For example, MCDM techniques can be
adapted to address specific challenges in efficiency analysis, such as
handling qualitative conditions or integrating stakeholder preferences into
efficiency models.

Further research could explore developing integrated models that leverage the
strengths of both MCDM and EA, tailored to specific sectors such as
healthcare, energy, and public services. Methodological innovations in
handling uncertainty, dynamic Conditions, and the inclusion of sustainability
dimensions are promising

Table 3: Solutions in Measuring the Super-Efficiencies of DMU-B and DMU-D. | ’DMU-o=’ |  | B | B | B | B | D | D | D | D
---|---|---|---|---|---|---|---|---|---|---
Row | Solution | Unit | sPT-II | sTS1-II | sTSz-II | sTS2-II | sPT-II | sTS1-II | sTSz-II | sTS2-II
R1 | $\kappa_{o}^{c}$ | - | 0.2417 | 0.2417 | 0.3345 | 0.4273 | 1.3411 | 1.3411 | 1.4092 | 1.4774
| $\tau_{o}$ | $ | 0.5 | 0.5 | 0.4823 | 0.4591 | 0.7709 | 0.8037 | 0.7827 | 0.7614
| $\Delta_{o},\delta_{o}$ | $ | 0.5855 | 0.5855 | 0.5964 | 0.5979 | 0.2611 | 0.2722 | 0.2688 | 0.2652
R2 | $v_{1o}$ | $/ton | 0 | 0 | 0 | 0 | 0.3889 | 0.4230 | 0.4119 | 0.4007
| $v_{2o}$ | $/hr | 0.0143 | 0.0172 | 0.0166 | 0.0158 | 0 | 0 | 0 | 0
| $u_{1o}$ |  $/ $m^{3}$ | 0.0009 | 0.0009 | 0.0009 | 0.0008 | 0.0003 | 0.0003 | 0.0003 | 0.0003
| $u_{2o}$ | $/% | 0.0056 | 0.0056 | 0.0054 | 0.0052 | 0.0024 | 0.0020 | 0.0020 | 0.0019
| $w_{o}$ | $ | 0.0000 | 0.3538 | 0.3413 | 0.3249 | 0 | 0.0566 | 0.0551 | 0.0536
R3 | $Q_{1o}$ | - | 0 | 0 | 0 | 0 | 0 | 0.0000 | 0.1156 | 0.2313
| $Q_{2o}$ | - | 0 | 0 | 0.3840 | 0.7681 | 0 | 0 | 0 | 0
| $P_{1o}$ | - | 0.4344 | 0.4344 | 0.2172 | 0.0000 | 0.3387 | 0.3387 | 0.2278 | 0.1170
| $P_{2o}$ | - | 0.7366 | 0.7366 | 0.6355 | 0.5343 | 0 | 0 | 0 | 0
| $\pi_{A}$ | - | 0.2417 | 0.2417 | 0.3345 | 0.4273 | 0 | 0 | 0 | 0
| $\pi_{B}$ | - | 0 | 0 | 0 | 0 | 0.6424 | 0.6424 | 0.5211 | 0.3997
| $\pi_{G}$ | - | 0 | 0 | 0 | 0 | 0.6986 | 0.6986 | 0.8881 | 1.0776
R4 | $\widehat{x}_{1o}$ | ton | 1 | 1 | 1 | 51.273 | 1.9 | 1.9 | 2.119 | 2.339
| $\widehat{x}_{2o}$ | hr | 29 | 29 | 40.137 | 567 | 281 | 281 | 281 | 281
| $\widehat{y}_{1o}$ | $m^{3}$ | 320.7 | 320.7 | 443.8 | 41.4 | 1617.6 | 1617.6 | 1888.8 | 2159.9
| $\widehat{y}_{2o}$ | % | 23.442 | 23.442 | 32.444 | 0.5897 | 97 | 97 | 97 | 97
| $\gamma_{o}$ | - | 0 | 0 | 0.3105 | 0.1388 | 0 | 0.0000 | 0.3367 | 0.6642
| $\omega_{o}^{a}$ | $ | 0 | 0.0855 | 0.1142 | 0 | 0 | 0.0759 | 0.0776 | 0.0792
R5 | $x_{1o}v_{1o}$ | $ | 0 | 0 | 0 | 0.4591 | 0.7389 | 0.8037 | 0.7827 | 0.7614
| $x_{2o}v_{2o}$ | $ | 0.4145 | 0.5 | 0.4823 | 0.4591 | 0 | 0 | 0 | 0
| $y_{1o}u_{1o}$ | $ | 0.5 | 0.5 | 0.4823 | 0.4591 | 0.7709 | 0.8037 | 0.7827 | 0.7614
| $y_{2o}u_{2o}$ | $ | 0.5 | 0.5 | 0.4823 | 0.4021 | 0.2291 | 0.1963 | 0.1912 | 0.1860
R6 | $\alpha_{o},\alpha_{o}^{c}$ | $ | 0.4145 | 0.4145 | 0.4036 | 1 | 0.7389 | 0.7278 | 0.7312 | 0.7348
| $\beta_{o},\beta_{o}^{c}$ | $ | 1 | 1 | 1 | 0.7547 | 1 | 1 | 1 | 1
| $\widehat{\beta}_{o},\widehat{\beta}_{o}^{c}$ | $ | 0.4145 | 0.4145 | 0.5888 | 0.7547 | 0.7389 | 0.7278 | 0.8217 | 0.9109
| $\widehat{\beta}_{o},\widehat{\beta}_{o}^{c}$ | $ | 0.4145 | 0.4145 | 0.5888 | 0.4021 | 0.7389 | 0.7278 | 0.8217 | 0.9109
R7 | $\Xi_{o}$ | - | 0.4145 | 0.4145 | 0.4036 | 2.4868 | 0.7389 | 0.7278 | 0.7312 | 0.7348
| $E_{o}$ | - | 2.4126 | 2.4126 | 2.4779 | 2.1621 | 1.3533 | 1.3740 | 1.3676 | 1.3609
R8 | $\alpha_{K},\alpha_{K}^{c}$ | $ | 2.0725 | 2.1462 | 2.1761 | 1.7663 | 0.6223 | 0.6202 | 0.6226 | 0.6232
| $\alpha_{A},\alpha_{A}^{c}$ | $ | 1.7151 | 1.7151 | 1.7603 | 0.4021 | 0.8945 | 0.9163 | 0.9109 | 0.9037
| $\alpha_{B},\alpha_{B}^{c}$ | $ | 0.4145 | 0.4145 | 0.4036 | 4.3150 | 0.3889 | 0.3664 | 0.3754 | 0.3827
| $\alpha_{D},\alpha_{D}^{c}$ | $ | 4.0163 | 4.4910 | 4.4378 | 3.8242 | 0.7389 | 0.7278 | 0.7312 | 0.7348
| $\alpha_{G},\alpha_{G}^{c}$ | $ | 3.5732 | 3.9565 | 3.9222 | 1.4497 | 0.7000 | 0.7048 | 0.7049 | 0.7033
| $\alpha_{H},\alpha_{H}^{c}$ | $ | 1.4293 | 1.3703 | 1.4277 | 1.2831 | 0.9723 | 1.0009 | 0.9933 | 0.9839
R9 | $\beta_{K},\beta_{K}^{c}$ | $ | 1.1889 | 1.1889 | 1.2527 | 1.7663 | 0.4422 | 0.4396 | 0.4466 | 0.4520
| $\beta_{A},\beta_{A}^{c}$ | $ | 1.7151 | 1.7151 | 1.7603 | 1 | 0.6473 | 0.6323 | 0.6343 | 0.6347
| $\beta_{B},\beta_{B}^{c}$ | $ | 1 | 1 | 1 | 2.6723 | 0.3889 | 0.3664 | 0.3754 | 0.3827
| $\beta_{D},\beta_{D}^{c}$ | $ | 2.702 | 2.702 | 2.712 | 1.938 | 1 | 1 | 1 | 1
| $\beta_{G},\beta_{G}^{c}$ | $ | 1.902 | 1.902 | 1.941 | 1.362 | 0.7000 | 0.7048 | 0.7049 | 0.7033
| $\beta_{H},\beta_{H}^{c}$ | $ | 1.2751 | 1.2751 | 1.3359 | 0 | 0.4805 | 0.4702 | 0.4765 | 0.4811
R10 | AP(x) | $ | 0 | -0.0855 | -0.0787 | 0.0000 | 0 | -0.076 | -0.051 | -0.027
| AP(y) | $ | 0 | 0 | 0.0354 | 0.1313 | 0 | 0 | 0.026 | 0.053

areas that can enhance the applicability and impact of these decision-support
tools. In conclusion, the interplay between MCDM methods and EA represents
fertile ground for advancing sophisticated and practical decision-making
frameworks, offering significant potential for improving system performance
and achieving strategic goals across diverse contexts.

The most effective way to help the decision-making unit become the best
performer is by conducting the structured four-phase procedure depicted in
Figure 1 that accurately estimates the necessary and achievable input and
output adjustments. This procedure is iterative and may require refinement as
new data become available or the operational context of the DMUs changes.

Theorems 2.1 to 4.3 have proved that DMU-o is comprehensively assessed. DMU-o
will mimic the best peers. Collaboration with domain experts and stakeholders
is crucial throughout the process to ensure the relevance and applicability of
the findings and recommendations.

The Simplex method [Dantzig and Thapa][9] is the backbone of the VGA models,
enabling post-optimality analysis and flexibility in evaluating DMU-o. This
analysis allows for adjustments based on different coefficients
$\textit{(}X,Y)$, which is crucial in real-world evaluations.

The adjustment ratios of inputs ($Q_{o}^{\star}$) typically range between 0
and 1, but outputs ($P_{o}^{\star}$) may exceed 1. Imposing limits like
$P_{r}\leq 1$ or setting upper and lower bounds, $\underline{Q}_{o}\leq
Q_{o}\leq\bar{Q}_{o}$ and $\underline{P}_{o}\leq P_{o}\leq\bar{P}_{o}$, to
meet its particular requirements.

Panwar et al.[10] reviewed the practical issues that have been addressed in
the DEA literature, such as the Malmquist Index, network DEA, free-disposal
hull, imprecise data, data mixed with 0s and negatives [Sueyoshi and Goto][11]
input/output sharing, and input/output selection [Ali and Seiford][12]. The
current best practice of the four-phase VGA procedure could be used in
specific studies to evaluate performance under the worst practice.

## Acknowledgment

We thank the referees for their valuable comments. This research received no
specific grant from public, commercial, or not-for-profit funding agencies. We
have no competing interests to declare. We confirm that the data supporting
the findings of this study are available within the article.

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##  Appendix A CRS and VRS DEA Models

###  A.1 DEA Theories.

Charnes et al.[13] introduced the concept of Data Envelopment Analysis (DEA),
which evaluates a set of entities known as decision-making units (DMUs). Each
DMU operates a system that consumes multiple inputs to produce outputs. The
observed data from these DMUs form a production possibility set (PPS) with an
efficiency frontier, where each DMU-o $(x_{o},y_{o})$ belongs to the PPS. The
PPS possesses four properties, denoted as (A1)–(A4)[Chapter 3, Cooper et
al.][4]. Banker et al.[14] and Banker et al.[15] introduced the convexity
condition to the PPS. Additionally, Chapters 1 and 2 in Sickles and
Zelenyuk[5] provided twenty graphical illustrations of the PPS in two
dimensions to underpin the theoretical foundations of CRS and VRS DEA models.

Eqs. (3.1) and (4.1) in Cooper et al.[4], as shown in Eqs. (94) and (95)
represent the PPS without and with the convexity condition, respectively,
under the constant returns-to-scale (CRS) and variable returns-to-scale (VRS)
assumptions.

$P^{CRS}=\\{(x_{o},y_{o})|x_{o}\leq X\lambda,y_{o}\geq Y\lambda,\lambda\geq
0.\\}$ (94) $P^{VRS}=\\{(x_{o},y_{o})|x_{o}\leq X\lambda,y_{o}\geq
Y\lambda,(e\lambda=1,)\lambda\geq 0.\\}$ (95)

Cooper et al.[4] offered the envelopment and multiplier programs of the
Additive Model, Eqs. (4.34)–(4.38) and Eqs. (4.39)–(4.43), to estimate the
vector of variable intensities $\lambda$. Halická and Trnovská[16] extensively
reviewed DEA models and proposed a unified model to encompass various DEA
approaches that address the Additive Model.

###  A.2 Use the Additive DEA Model to Solve the CRS and VRS PPS.

In the following Eqs. (96)–(100) and Eqs. (101)–(104) depicts the envelopment
and multiplier program of the unified DEA model. Including and excluding the
parenthesized elements, the program represents the VRS and CRS measurement. In
which, Eqs. (97)–(100) constitute Eqs. (94) and (95).

The Additive Model measures the inefficiency of the object DMU, DMU-o, to
obtain efficient peers in its regards. Every DMU in the set of DMUs takes
turns to be DMU-o to constitute the entire efficiency frontier of the PPS.

####  A.2.1 The Envelopment (primal) Program.

$F_{o}^{\star}=\max_{\lambda_{o}\textbf{,}\textbf{s}_{o}^{x},\textbf{s}_{o}^{y}}\sum_{\forall
i\in I}s_{io}^{x}b_{i}^{x}+\sum_{\forall r\in R}s_{ro}^{y}b_{r}^{y},\forall
o\in J;$ (96) $s.t.\sum_{\forall j\in
J}x_{ij}\lambda_{oj}+s_{io}^{x}=x_{io},\forall i\in I;$ (97) $-\sum_{\forall
j\in J}y_{rj}\lambda_{oj}+s_{ro}^{y}=-y_{ro},\forall r\in R;$ (98)
$(\sum_{\forall j\in J}\lambda_{oj}=1;)$ (99) $\lambda_{oj},\forall j\in
J;s_{io}^{x},\forall i\in I;s_{ro}^{y},\forall r\in R\geq 0.$ (100)

Each DEA model uses the variable slacks of input-i and output-r, $s_{io}^{x}$
and $s_{ro}^{y}$, to ensure that DMU-o projects on the efficiency frontiers of
the two PPSs. The coefficients $b_{i}^{x}$ and $b_{r}^{y}$ are artificial goal
weights corresponding to input-i and output-r. Halická and Trnovská[16] listed
objective functions and artificial goal weights used in several DEA models in
Tables 2 and 3.

####  A.2.2 The Multiplier (dual) Program.

The envelopment program can be converted to the multiplier program, Eqs.
(101)–(104). Usually, the multiplier program is not presented when introducing
a DEA model, and the duality characteristics of the model are not verified. As
a result, the solutions of the multiplier model are not examined.

$\displaystyle
f_{o}^{\star}=\min_{\textbf{v}_{o},\textbf{u}_{o},\sigma_{o}}\sum_{\forall
i\in I}v_{io}x_{io}-\sum_{\forall r\in R}u_{ro}y_{ro}(+1\sigma_{o}),\forall
o\in J;$ (101) $s.t.\sum_{\forall i\in I}v_{io}x_{ij}-\sum_{\forall r\in
R}u_{ro}y_{rj}(+1\sigma_{o})\geq 0,\forall j\in J;$ (102) $v_{io}\geq
b_{i}^{x},\forall i\in I;u_{ro}\geq b_{r}^{y},\forall r\in R;$ (103)
$v_{io},\forall i\in I;u_{ro},\forall r\in R(,\sigma_{o})free.$ (104)

The inefficiency condition Eq. (102) of each DMU-j bounds its excess at a
minimum of zero and corresponds to a variable intensity $\lambda_{oj}$ in the
primal program. The model identifies a set of efficient reference peers with
zero inefficiency score and positive intensity, while other DMUs are deemed
inefficient with zero intensity. The weighting condition Eq. (103) of an input
(or output) corresponds to a variable slack, $s_{io}^{x}$ (or $s_{ro}^{y}$),
in the primal program, and restricts its variable weight, $v_{io}$ (or
$u_{ro}$), to a lower bound defined by a specified artificial goal weight,
$b_{i}^{x}$ (or $b_{r}^{y}$).

###  A.3 Drawbacks of the Envelopment Program.

####  A.3.1 Efficiency Frontier of the PPS is not Identified.

Decision variables in the VRS PPS Eq. (95), $\lambda_{oj},\forall j\in$ J, are
estimated by the above envelopment program. Hyperplanes Eqs. (97)–(100)
construct the convex feasible space of the envelopment program. The optimal
extreme point is located at one of the extreme points of the convex feasible
space: the intersection of the objective function and several hyperplanes. On
the other hand, DMU-o has a project point on the PPS frontier that is
constituted by efficient DMUs.

The optimal extreme point and projection point belong to the convex feasible
space and PPS space. These two points are identified by the decision variables
and observed data. The envelopment program cannot identify the projection
point because artificial goal weights are still under research. Hence, the
estimated intensities and slacks could not be applied to the projection point
on the PPS. Similarly, the data envelopment program cannot solve the CRS PPS
Eq. (94).

####  A.3.2 Incomplete Solutions of the Envelopment Program.

Each benchmark condition Eqs. (97) and (98) corresponds to a variable weight,
$v_{io}$ (or $u_{ro}$), in the dual (multiplier) program. Using the artificial
goal weights Eq. (96) maximizes the inefficiency score of DMU-o, $F_{o}^{*}$.
Optimized solutions for envelopment and multiplier programs are used to verify
their correctness. As described in the next section, the multiplier program
obtains incomplete solutions. Hence, envelopment solutions may be incomplete.

####  A.3.3 The Envelopment Program Does not have Convexity Condition.

Both CRS and VRS envelopment programs have convex feasible spaces. The
”convexity condition” Eq. (99) reduces the space of the CRS envelopment
program. It is not equivalent to restricting the non-convex CRS PPS space Eq.
(94) to have the convex VRS PPS space Eq. (95). Eq. (99) has a particular
scalar 1. In our VGA models, each DMU-j has an intensity $\pi_{oj}$, $\forall
j\in J$. DMU-o managing the SIC scalar $\kappa_{o}^{c}$ is significant in the
efficiency measurements, where SIC is the condition $\sum_{\forall j\in
J}\pi_{oj}=\kappa_{o}^{c}$.

###  A.4 Drawbacks of the Multiplier Program.

####  A.4.1 The Unrealistic Reference Peers.

The set of artificial goal weights, Eq. (103), is applied to evaluate every
DMU-o, regardless of its configurations of inputs and outputs. The identified
reference peers may not have analogous configurations with DMU-o. The
optimized solutions are unrealistic.

####  A.4.2 Inaccurate Solutions of CRS models.

In the CRS model, DMU-o has an efficiency score equal to the ratio between the
weighted sum of outputs and the weighted sum of inputs, see Eq. (11) in
[Banker et al.][14], $\sum_{\forall r\in R}u_{ro}y_{ro}/\sum_{\forall i\in
I}v_{io}x_{io}$. Eq. (101) aims to minimize DMU-o’s excess $f_{o}^{*}$, the
inefficiency score (=1-efficiency score). The model can produce inaccurate
solutions when the estimated virtual input $\sum_{\forall i\in
I}v_{io}^{*}x_{io}$ is not equal to 1. CRS DEA models use various artificial
goal weights and cannot ensure the estimated virtual input equals 1.

####  A.4.3 Inaccurate Solutions of VRS models.

In Eqs. (101) and (102) of the VRS model, the inefficiencies scores are
affected by $1\sigma_{o}$, without theoretical support. However, Halická and
Trnovská[16] Section 4 concluded that VRS models suffer from duality issues
and unbounded solutions. The inefficiency score $f_{o}^{*}$ cannot be
obtained.

####  A.4.4 Mistakenly Using Dimensionless Measurement

Banker et al.[14] used Figure 2 to depict the particular VRS PPS with one
input (x) and one output (y). The line segments passing through points A and E
indicate the increasing and decreasing returns to scale by the sign of
$\sigma_{o}$. Each line respectively intercepts with the vertical and
horizontal axis with a length of -$\sigma_{o}$ and has the same measurement
unit of x and y. In this case, Eq. (101) is expressed as following:

$f_{o}^{\star}=\min_{v_{o},u_{o},\sigma_{o}}v_{o}x_{o}-u_{o}y_{o}+1\sigma_{o}.$
(105)

In the above equation, the four elements,
$f_{o}^{\star},v_{o}x_{o},u_{o}y_{o}$, and $1\sigma_{o}$, should have been
measured by the virtual currency $. Hence, $f_{o}$ is not the dimensionless
inefficiency score. Our VGA models have fixed this problem for cases with
multiple inputs and outputs.

####  A.4.5 Returns to Scale Is not Measured.

The actual measurement of the returns to scale in practice relies on the
concept of scale elasticity and scale efficiency, see Page 31 in Sickles and
Zelenyuk[5]. However, no one in the DEA literature has presented a solution
for multiple inputs and outputs.

###  A.5 Cannot Measure Super-efficiencies.

The CRS and VRS ARRs DEA models identified Pareto-efficient DMUs with an
efficiency score equal to 1. These models are modified to estimate each
efficient DMU’s super-efficiency, which is more significant than 1. A DMU
exhibiting higher super-efficiency surpasses other efficient DMUs. An
efficient DMU expands and reduces inputs and outputs, deteriorating its super-
efficiency to 1. Our second scenario furnishes super-efficiencies, including
and excluding the SIC condition.

Chen and Du[17] reviewed linear, nonlinear, and integer VRS models, which
attempted to measure super-efficiencies. They concluded that convexity
constraints caused the infeasibility issue. Unfortunately, those linear
programming-based models discussed in [Chen and Du][17] only present the
envelopment program without the multiplier program; those models may not
fulfill the duality characteristics. The CRS models have solutions of the
envelopment program but without the multiplier program’s solutions. CRS and
VRS DEA models have the same problem in measuring inefficiencies and super-
efficiencies, even if zero data does not exist.

###  A.6 Heterogeneity Caused Unrealistic Evaluations.

Li et al.[18] raised concerns regarding input configurations significantly
differing among DMUs. Aleskerov and Petrushchenko[19] argued that DEA may not
be applicable when DMUs operate under vastly different conditions and have
significantly distinct input and output compositions. This heterogeneity poses
challenges for analysis.

DEA models rely on artificial goal weights and consequently identify reference
peers with non-analogous criteria configurations relative to DMU-o. Instead,
we propose that DMU-o systematically determines the goal price of virtual
inputs and outputs to identify reference peers with analogous criteria
configurations.

###  A.7 Summaries

This subsection summarizes foundational research and highlights critical
limitations for applying DEA models in our study.

Using linear programming frameworks is a notable strength of DEA models,
allowing for robust efficiency analyses. However, despite the extensive
exploration of artificial goal weights within DEA theories, several critical
drawbacks remain:

The solutions provided are incomplete, leaving gaps in the efficiency
analysis.

DMU-o struggles to identify reference peers with analogous criteria
configurations, resulting in unrealistic estimated criteria benchmarks.

DMU-o fails to select achievable benchmarks and thus cannot attain a realistic
efficiency score.

The ranking of Pareto-efficient DMUs according to their super-efficiencies is
not feasible.

Modified and extended Additive Models, utilized in various applications,
suffer from similar drawbacks and have not been thoroughly discussed in the
literature. Researchers such as Dyson et al.[20], Cooper et. al.[4],
Brown[21], Zarrin et al.[22], and Zhu et al.[23] have extensively examined the
limitations and methodologies of DEA. Subsequent literature reviews by Mergoni
and Witte[24] and Panwar et al.[8] further emphasize the ongoing challenges
within DEA models despite numerous refinements. Additionally, Emrouznejad and
Yang[25] highlighted the extensive scope of DEA literature, noting the
existence of over ten thousand journal articles and a hundred textbooks
published in the past fifty years.

These summaries underscore the persistent challenges and the extensive
research efforts aimed at addressing the limitations of DEA models, which are
crucial for the methodological framework of our study.

##  Appendix B Comparisons Between DEA and VGA

Table 4 summarizes the comparisons of the first scenario, with a footnote
providing additional references for further reading. Specifically, DEA models
focus on inefficiency scores, while VGA models target virtual gaps. It is
noted at the end of the table that DEA’s inefficiency scores can be infeasible
and unbounded, posing a significant limitation. In contrast, VGA models
generate feasible inefficiency scores, effectively addressing this issue. The
DMU-o selects the SIC scalar to ensure that these evaluations and benchmarks
are practical and attainable.

Charnes et al.[13] and Banker et al.[14] introduced the CRS and VRS DEA models
and proposed a two-phase DEA procedure to assess DMU-o. In Phase One, the
method obtains overestimated one-sided radial efficiency scores. In Phase Two,
the Additive Models measure the remaining inefficiencies. However, the optimal
solutions obtained from the two phases cannot be aggregated for further
applications.

In contrast, VGA models PT and TSc, which utilize the specific SIC scalar
$\kappa_{o}^{c}=1$, provide the lower bound efficiency scores of the CRS and
VRS DEA models. As illustrated in Appendix A, while CRS and VRS Additive
Models offer a robust theoretical basis, they often result in incomplete
solutions, highlighting their limitations in practical applications. These
models form the foundational frameworks of all DEA models and are essential
for understanding efficiency measurements. The first and second VGA models
enhance these frameworks by addressing their shortcomings and providing a more
detailed and holistic evaluation of efficiency.

##  Appendix C  Introduction to Computational Tools

In our study, the linear programming (LP) models showcased in Tables 2 and
LABEL:table:3 were solved using Solver, an Add-In for Microsoft Office Excel.
This tool facilitates the optimization of models by adjusting the values of
the decision variables to meet the constraints and objectives specified by the
user. We have meticulously structured the LP models within the accompanying
Excel file to enhance understanding and facilitate replication. Each model is
comprehensively laid out, showcasing the parameters $(X,Y)$ and
$(x_{o},y_{o})$, decision variables, and constraints.

The functionality embedded within each solution symbol is detailed, allowing
readers to trace the computation logic seamlessly. For clarity, consider the
following example from Table 3, DMU-K Evaluation. This evaluation involves
models PT-K, TS1-K, and TS2-K. The scalar $\kappa_{o}^{1}$, computed in the
PT-K model, is a crucial link, hyperlinked to the TS1-K model for further
computation. Similarly, the ”Sensitivity Report TS1-K” spreadsheet elucidates
the calculation of $\kappa_{o}^{2}$, which, in turn, is hyperlinked to the
TS2-K model. This interconnected setup not only facilitates a deeper
understanding of the computational process but also aids in the transparent
communication of how each model’s outputs contribute to subsequent
calculations.

To ensure the utility of the Excel models, each spreadsheet is designed with
readability and ease of navigation in mind. Hyperlinks between related models
and their components encourage an intuitive exploration of the computational
logic underpinning our research findings. The computations for the second
scenario are similar.

Table 4: Comparing DEA vs. VGA Models.

Row |  Compare | Model | Model
---|---|---|---
1 |  |  CRS DEA |  First VGA |  VRS DEA |  Second VGA
2 |  Decision variables |  Criteria Weights(1/†) |  Criteria virtual unit prices ($) |  Criteria Weights (1/†) |  Criteria virtual unit prices ($)
3 |  |  Weighted criteria (-) |  Criteria virtual prices ($) |  Weighted criteria (-) |  Criteria virtual prices ($)
4 |  Lower bounds |  An artificial goal weight for each weight (1/†) |  A unified Systematic goal price for all virtual prices ($) |  An artificial goal weight for each weight (1/†) |  A unified Systematic goal price for all virtual prices ($)
5 |  Decision variables |  Criteria Slacks(†) |  Criteria adjustment ratios (-) |  Criteria Slacks(†) |  Criteria adjustment ratios (-)
6 |  Decision variables |  Intensities of DMUs (-)‡1 |  Intensities of DMUs (-) ‡2 |  Intensities of DMUs (-) ‡3 |  Intensities of DMUs (-) ‡4
7 |  Decision variable |  |  |  SIC Scalar weight (-) ‡5 |  SIC Scalar unit price ($) ‡6
|  Virtual scalar |  |  |  1$\times$ SIC Scalar weight (-) |  Scalar $\times$ SIC Scalar unit price ($)
8 |  Objective function |  Inefficiency score $F_{o}^{*}$ and $f_{o}^{*}$ (-) |  Virtual gap ($) |  Inefficiency score $F_{o}^{*}$ and $f_{o}^{*}$ (-) |  Virtual gap ($)
9 |  Objective values |  ${}_{o}^{*}$= total of products of slacks and artificial goal weights (-) |  The total adjustment ratios multiply the systematic goal price ($) |  $F_{o}^{*}$= total of products of slacks and artificial goal weights (-) |  The total adjustment ratios multiply the systematic goal price ($)
10 |  Objective values |  $F_{o}^{*}$= virtual input minus output (-) |  Virtual input minus output ($) |  $F_{o}^{*}$= virtual input minus output plus scalar weight (-) |  Virtual input minus output plus (virtual scalar)= (Affected virtual input)-(Affected virtual Output) ($) ‡7
11 |  0 $<F_{o}^{*}\leq 1$ |  Not satisfied |  $F_{o}^{*}$ = (Virtual gap)/(Virtual input) |  Unbounded solutions ‡8 |  $F_{o}^{*}$=1-(Affected virtual output) / (Affected virtual input)
12 |  Reference peers |  Non-analogous with DMU-o |  Analogous with DMU-o |  Not solved |  Analogous with DMU-o
13 |  SIC scalar |  Not Available |  Not Available |  =1 ‡9 |  Within a systematic range ‡10
14 |  Scalar selection |  No |  No |  Not allowed |  Allowable‡11
15 |  Benchmarks of inputs & outputs |  Unrealistic |  Realistic |  Not solved |  Realistic & achievable

Note: Measurement units are indicated in the parenthesize such as (-)
dimensionless, ($) the virtual currency $, and (†) the same as the criterion.
‡1, ‡3, and ‡5: see [Eqs. (3.1), (4.38), and (4.43)] in Cooper et al.[4]; ‡2,
‡4, and ‡6 are illustrated in Section 2.1; Section 2.3 derives ‡7. ‡8: see
Halická and Trnovská[16]. ‡9 is digested from [Eqs. (4.1) & (4.37)] of Cooper
et al.[4]. ‡10 and ‡11 can be find in Section 4.