Performance measurement of decision making units through interval efficiency with slacks-based measure: an application to tourist hotels in Taipei

Abstract

Data envelopment analysis (DEA) is widely used to evaluate the performance of a group of homogeneous decision making units (DMUs). Considering the uncertainty, interval DEA has been introduced to fit into more situations. In this paper, an interval efficiency method based on slacks-based measure is proposed to solve the uncertain problems in DEA. Firstly, the maximum and minimum efficiency values of the evaluated DMU are calculated by the furthest and closest distance from the evaluated DMU to the projection points on the Pareto-efficient frontier, respectively. Then, the AHP method is used for the full ranking of DMUs. The paper uses the pairwise comparison relationship between each pair of DMUs to construct the interval multiplicative preference relations (IMPRs) matrix. If the matrix does not meet the consistency condition, a method to obtain consistency IMPRs is introduced. According to the consistency judgment matrix, the full ranking of DMUs can be obtained. Finally, we apply our method to the performance evaluation of 12 tourist hotels in Taipei in 2019.

Keywords

Performance measurement data envelopment analysis interval efficiency interval multiplicative preference relations full ranking

1 Introduction

Data envelopment analysis (DEA) is a data-driven tool for performance evaluation of a set of homogenous decision making units (DMUs) especially with multiple inputs and multiple outputs. Since Charnes et al. [1] proposed the Charnes-Cooper-Rhodes (CCR) model, DEA has attracted attentions from many scholars. Banker et al. [2] extended traditional DEA to Banker-Chanes-Cooper (BBC) model which is under the assumption of variable returns to scale. After that, many scholars proposed different DEA models, such as the super-efficiency DEA model [3], integer DEA model [4], additive DEA model [5], guaranteed domain DEA model [6], cross efficiency DEA model [7]. In the past 40 years, a series of expanded DEA models that extend the DEA theory and methods have been proposed. At the same time, some scholars applied the DEA method flexibly to empirical research, which promoted the application of DEA in practical production activities and management practice. At present, DEA has been widely used in various fields, such as college teaching performance evaluation [8], bank performance evaluation [9, 10], environmental performance assessment [11, 12], and supply chain sustainabilityassessment [13].

Traditional DEA methods do not take into account the preferences of decision-makers and the uncertainty of input-output data [14]. In the actual situation, there are a lot of uncertainty factors. On the one hand, since decision-makers have different preferences, the expected goals they want to achieve are variable. On the other hand, DMUs’ performances often cannot be accurately measured due to the existence of some inevitable influencing factors on the input-output data. In this case, the traditional DEA models do not conform to the actual situation and cannot accurately and effectively evaluate the relative efficiency of DMUs. Therefore, some scholars extend the traditional DEA model to the interval DEA model, and the efficiency obtained by the interval DEA better considers the uncertain factors rather than traditional DEA in reality.

Based on traditional radial DEA models, Entani et al. [15] proposed the interval DEA based on the radial CCR model. They evaluated DMUs’ performances from the perspective of optimism and pessimism respectively, and the obtained efficiency score is an interval number. Amirteimoori et al. [16] also determined a cost-efficiency interval from both optimistic and pessimistic perspectives. In their paper, the efficiency from the optimistic perspective is the upper bound of the cost efficiency interval, and the efficiency from the pessimistic perspective is the lower bound of the cost efficiency interval. However, Azizi [17] pointed out that the model of Entanni et al. [15] has a major flaw in calculating DMUs’ pessimistic efficiencies, that is, this method only considers the data of one input and one output of the evaluated DMUs and ignores the rest of the input-output data. On this basis, Azizi [18] proposed a new interval DEA model based on the concepts of optimistic and pessimistic efficiency. Compared with the model of Entani [15], this model took into account all input-output data when measuring the pessimistic efficiencies of DMUs. Based on Doyle and Green [19], Liu [20] proposed an interval cross-efficiency method by considering cross-efficiency intervals which is formed by using the aggressive and benevolent strategy in cross efficiency evaluation. An et al. [21] provided an approach to derive the maximum and minimum cross efficiencies by using the benevolent and aggressive strategy. Yang et al. [22] proposed an alternative strategy that does not consider the preference of the decision maker in choosing aggressive or benevolent strategy. They considered all possible weight sets in weight space when computing the cross efficiency and each DMU is given an interval cross efficiency. With the derived interval cross-efficiency matrix for all DMUs by the above strategy, Ang et al. [23] determined cross-efficiency intervals, ranking ranges, performance maverick and diversity for each DMU, which could provide more insightful information in cross-efficiency evaluation for decision makers.

However, the previous interval DEA models are applicable only when the input-output can be improved proportionately because they just solved the uncertainty of performance evaluation in radial DEA models. In the non-radial DEA, there is also the problem of inaccurate efficiency measurement results due to the uncertainty of decision-makers’ preferences. Tone [24] proposed a slacks-based measure (SBM) model to calculate efficiency where the input-output data project to the strongly efficient frontier disproportionately. In contrast to CCR and BBC models, the SBM model deals directly with input excess and output shortfall. The SBM model has some desirable properties, such as unit invariant, strong monotonic and reference-set dependent. However, it suffers from one drawback, that is, it only considers the furthest target from the strongly efficient frontier, and thus the efficiency measure obtained may not conform to the perspective of decision-makers in practice.

Another issue in interval DEA is the performance ranking of DMUs. After the interval efficiency is obtained, how to complete the ranking of DMUs is a crucial problem. Some scholars put forward the AHP/DEA model to full rank all DMUs. Mirhedayatian and Saen [25] proposed a modified DEA/AHP method to generate logical weights consistent with decision-makers judgments. However, the proposed method retained the subjectivity of the judgment matrix in AHP [26, 27]. For this reason, Sinuany-Stern et al. [28] proposed a two-stage AHP/DEA sorting model, combining DEA and AHP. They first used DEA to calculate the efficiency score of DMUs, and then applied the preference relationship to the total ranking of DMUs. In their method, the preference relation that forms the judgment matrix is derived from the comparison of efficiency value and does not involve the utility preference of the decision-maker, so it overcomes the disadvantage of the analytic hierarchy process. On this basis, An et al. [21] combined DEA and AHP to complete the ranking of DMUs according to all possible cross efficiencies of DMUs relative to other DMUs.

To deal with the above issues, this paper proposes an interval efficiency method with slacks-based measure, under which the furthest and closest targets are considered. After obtaining the interval efficiency, an AHP/DEA method is proposed to process the performance ranking of DMUs. Firstly, the SBM model proposed by Tone [24] is used to calculate the maximum distance to the strongly efficient frontier, and the minimum distance to the strongly efficient frontier is derived through the closest target model [29]. Through the above two models, we can calculate the furthest and closest distances of the evaluated DMU to the efficient frontier respectively, and further obtain the maximum and minimum efficiency scores of the evaluated DMU. Thus, the possible efficiency interval of the evaluated DMU is obtained. Then, we combined DEA and AHP methods to realize the performance ranking of DMUs. Based on the interval efficiency, the pairwise comparison relationship between the efficiency of each pair of DMUs is used to construct the IMPRs matrix. If the matrix does not meet the consistency condition, this paper introduces a method to obtain consistency IMPRs. According to the consistency judgment matrix, the performance ranking of DMU can be obtained.

Our method inherits some advantages of previous interval DEA methods. 1) Since the DMUs’ performance evaluation can be obtained from different aspects, it is better to use efficiency as an interval to reflect performance rather than a single value. 2) The interval DEA is more detailed than the conventional DEA when analyzing the performance of DMUs because it includes all possible efficiency scores. 3) Compared with single values, interval DEA better considers the preference of decision-makers according to different targets setting. Furthermore, the interval DEA model proposed in this paper has the following contributions compared with the previous interval DEA method. 1) It evaluates efficiency with slacks-based measures, and it can deal directly with input excess and output shortfall. 2) It considers both the furthest and closest distance from the evaluated DMU to the Pareto-efficient frontier. 3) The AHP is used to process the ranking of interval efficiency.

The rest of this paper unfolds as follows. In the second section, we define the efficiency interval of the non-radial model according to the criteria of furthest and closest targets. Section 3 builds the relative efficiency pairwise comparison between each pair of DMUs, then constructs IMPRs. After that, we introduce a method to derive consistent IMPRs. In Section 4, we apply the proposed method to rank the performance of tourist hotels in Taipei. Section 5 summarizes the paper.

2 Efficiency intervals of the non-radial model

In this section, we introduce the method of getting the efficiency interval. The SBM model is used to calculate the maximum distance to the strongly efficient frontier, and the minimum distance the strongly efficient frontier is derived through the closest target model.

2.1 Maximum distance to strong efficient frontier through SBM model

Assume that there are n DMUs, which use m inputs to produce s outputs. Each DMU is denoted as DMU_j (j = 1, 2, …, n), the observed inputs and outputs can be denoted by x_j = (x_1j, x_2j, …, x_mj) ^T and y_j = (y_1j, y_2j, …, y_qj) ^T, respectively. We stipulate the input data and the output data are positive, i.e., x_j > 0 and y_j > 0. According to the SBM model proposed by Tone [24], the efficiency value of DMU_k can be calculated as follows: $E_{k}^{-} = \min \frac{1 - \frac{1}{m} \sum_{i = 1}^{m} \frac{s_{i}^{-}}{x_{ik}}}{1 + \frac{1}{q} \sum_{r = 1}^{q} \frac{s_{r}^{+}}{y_{rk}}}$ $s . t . \sum_{j = 1}^{n} x_{ij} λ_{j} + s_{i}^{-} = x_{ik}, i = 1, 2, \dots, m$ $\sum_{j = 1}^{n} y_{rj} λ_{j} - s_{i}^{+} = y_{rk}, r = 1, 2, \dots, q$ $s_{i}^{-} ⩾ 0, i = 1, 2, \dots, m$ $s_{r}^{+} ⩾ 0, r = 1, 2, \dots, q$ $λ_{j} ⩾ 0, j = 1, 2, \dots, n$ (1)

Using the Charnes - Cooper transformation (see [1]), the model above can be transformed into linear programming. Then through the SBM model, the efficiency value of DMU_k denoted by $E_{k}^{-}$ can be derived. If the efficiency value $E_{k}^{-}$ is equal to 1, it means that the evaluated DMU is strongly efficient. Slacks $s_{i}^{-}, s_{r}^{+}$ indicate the excess of the ith input and shortfall of the rth output, respectively. Among them, input and output inefficiency are reflected in: $\frac{1}{m} \sum_{i = 1}^{m} \frac{s_{i}^{-}}{x_{ik}} \frac{1}{q} \sum_{r = 1}^{q} \frac{s_{r}^{+}}{y_{rk}}$

Note that the objective function of the SBM model is to minimize the efficiency value, in other words, the inefficiency value of input and output is maximized. The projection point of the evaluated DMU is the furthest point from the evaluated DMU to the efficient frontier. In fact, SBM is unreasonable in some cases. For instance, from the view of the decision-maker, it is desirable to reach the efficient frontier with the shortest path in some cases (for example, benchmarking inefficient DMUs). So, we put forward the calculation method of the minimum distance to the strongly efficient frontier below.

2.2 Minimum distance to strong efficient frontier through closest target model

Opposite to the SBM model, the closest target model proposed by Aparicio [29] evaluates the efficiencies of DMUs by referring to the closest point on strong efficient frontier. So, the minimum distance to the strongly efficient frontier can be derived from the closest target model. According to the method proposed by Aparicio [29], the following fractional linear programming is solved to obtain the minimum distance to the strongly efficient frontier. $E_{k}^{+} = \max \frac{1 - \frac{1}{m} \sum_{i = 1}^{m} \frac{s_{i}^{-}}{x_{ik}}}{1 + \frac{1}{q} \sum_{r = 1}^{q} \frac{s_{i}^{+}}{y_{ik}}}$

$s . t . \sum_{j ɛ E} x_{ij} λ_{j} + s_{i}^{-} = x_{ik}, i = 1, 2, \dots, m$ (2.1)

$\sum_{j ɛ E} y_{rj} λ_{j} - s_{i}^{+} = y_{rk}, r = 1, 2, \dots, q$ (2.2)

$s_{i}^{-} ⩾ 0, i = 1, 2, \dots, m$ (2.3)

$s_{r}^{+} ⩾ 0, r = 1, 2, \dots, q$ (2.4)

$λ_{j} ⩾ 0, j ɛ E$ (2.5)

$\sum_{i = 1}^{m} ν_{i} x_{ij} + \sum_{r = 1}^{q} μ_{i} y_{rj} + d_{j} = 0, j ɛ E$ (2.6)

$ν_{i} ⩾ 1, i = 1, 2, \dots, m$ (2.7)

$μ_{i} ⩾ 1, r = 1, 2, \dots, q$ (2.8)

$d_{j} ⩽ M b_{j}, j ɛ E$ (2.9)

$λ_{j} ⩽ M (1 - b_{j}), j ɛ E$ (2.10)

$b_{j} ɛ {0, 1}, j ɛ E$ (2.11)

$d_{j} ⩾ 0, j ɛ E$ (2.12)

where M is a big positive quantity.

In model (2), the constraints (2.1) - (2.5) are equivalent to the constraints in model (1), constraints (2.6) - (2.8) are equal to the multiplier form of the additive model. By adding in the constraint multiplier model, we can limit the reference set is restricted to the same hyperplane. Finally, constraints (2.9) - (2.12) are the key conditions that connect the above two sets of constraints.

If b_j = 0, then d_j = 0. Constraint (2.6) is equivalent to $- \sum_{i = 1}^{m} ν_{i} x_{ij} + \sum_{r = 1}^{q} μ_{i} y_{rj} + d_{j} = 0, j ɛ E$ , and constraint (2.9) becomes λ_j ⩽ M, jɛE. At this point, the DMU_j is the reference benchmark for the DMU_k. If b_j = 1, then d_j ⩽ M, Constraint (2.6) is equivalent to $- \sum_{i = 1}^{m} ν_{i} x_{ij} + \sum_{r = 1}^{q} μ_{i} y_{rj} ⩽ 0, j ɛ E$ , and constraint (2.9) becomesλ_j = 0, jɛE. Thus, the DMU_j is not the reference benchmark for the DMU_k. Through model (2), we can obtain the efficiency of DMU_k, denoted as $E_{k}^{+}$ . $E_{k}^{+}$ represents the efficiency value when DMU_k projects to the closest reference points on the strong efficient frontier with the minimum distance.

2.3 Efficiency interval

Through model (1) and model (2), we calculate the furthest and closest distance from the evaluated DMU to the efficient frontier, respectively, so as to obtain the maximum and minimum efficiency values of the evaluated DMU. In this way, the minimum efficiency value of the evaluated DMU is $E_{k}^{-}$ and the maximum efficiency value is $E_{k}^{+}$ . Therefore, the interval efficiency of DMU_k can be denoted by [ $E_{k}^{-}, E_{k}^{+}$ ] in the current paper.

3 Fully rank DMUs using the AHP approach

By calculating the efficiency value of each DMU, DEA can easily classify DMUs into efficient and inefficient categories. However, the proposed DEA method is not sufficient to fully rank DMUs. In order to reveal the relationship among DMUs and derive a more reasonable ranking, this paper use the AHP approach to implement a full sort of DMUs. First, we need to construct the judgment matrix [30, 31]. In this paper, we construct the IMPRs because the efficiencies of DMUs are interval numbers. The following steps elaborate on the method of this paper.

3.1 Interval multiplicative preference relations

Based on the interval efficiency obtained from the last section, we can calculate the relative efficiency pairwise comparison between each pair of DMUs, and further construct the IMPRs matrix.

In order to deal with the problem of uncertainty, Saaty and Vargas [32] introduced the concept of IMPRs, which is shown in the following.

Definition 1 [32] Let X ={ x₁, x₂, …, x_n } be the set of compared objects, then a typical matrix of interval pairwise comparison can be given by $\bar{A} = {({\bar{a}}_{ij})}_{n \times n}$ $= (\begin{matrix} [1, 1] & [a_{12}^{-}, a_{12}^{+}] & . . . & [a_{1 n}^{-}, a_{1 n}^{+}] \\ [a_{21}^{-}, a_{21}^{+}] & [1, 1] & . . . & [a_{2 n}^{-}, a_{2 n}^{+}] \\ ⋮ & ⋮ & ⋱ & ⋮ \\ [a_{n 1}^{-}, a_{n 1}^{+}] & [a_{n 2}^{-}, a_{n 2}^{+}] & . . . & (1, 1) \end{matrix}),$

$\bar{A}$ is called IMPRs, where $a_{ij}^{-}, a_{ij}^{+} ⩾ 0$ and $a_{ij}^{-} ⩽ a_{ij}^{+}, a_{ij}^{-} = 1 / a_{ij}^{+}$ , in which ${\bar{a}}_{ij}$ indicate that x_i is between $a_{ij}^{-}$ and $a_{ij}^{+}$ times as important as x_j. If $a_{ij}^{-} = a_{ij}^{+}$ for all i, j = 1, 2, …, n, $\bar{A}$ degenerates to a reciprocal preference relation [33].

Let E_k denote the efficiency of DMU_k (k = 1, 2, …, n). E_k is bounded in the interval ${\bar{E}}_{k} = [E_{k}^{-}, E_{k}^{+}]$ according to Section 2. Then, the efficiency pairwise comparison between DMU_i and DMU_j can be expressed by $a_{ij} = \frac{E_{i}}{E_{j}} (i, j = 1, 2, \dots, n)$ (3)

$a_{ij} \in [E_{ij}^{-}, E_{ij}^{+}] = [\frac{E_{i}^{-}}{E_{j}^{+}}, \frac{E_{i}^{+}}{E_{j}^{-}}] (i, j = 1, 2, \dots, n)$ (4)

In the above formula, a_ij reflects the preferred degree of the DMU_i over DMU_j. Thus, the efficiency pairwise comparison between DMU_i and DMU_j can be denoted by ${\bar{E}}_{ij} = [E_{ij}^{-}, E_{ij}^{+}] = [\frac{E_{i}^{-}}{E_{j}^{+}}, \frac{E_{i}^{+}}{E_{j}^{-}}] (i, j = 1, 2, \dots, n)$ (5)

From the above formula, we can easily derive that the efficiency pairwise comparison between DMU_j and DMU_i is ${\bar{E}}_{ji} = [E_{ji}^{-}, E_{ji}^{+}] = [\frac{E_{j}^{-}}{E_{i}^{+}}, \frac{E_{j}^{+}}{E_{i}^{-}}] = \frac{1}{{\bar{E}}_{ij}}$ (6)

Note that when i = j, we can obtain that ${\bar{E}}_{ii} = [1, 1]$ for all i = 1, 2, …, n.

After getting the relative efficiency pairwise comparison between each pair of DMU, we can construct IMPRs. ${DMU}_{1} {DMU}_{2} . . {\dot{DMU}}_{n}$ (7) $\bar{E} = (\begin{matrix} [1, 1] & {\bar{E}}_{12} & . . . & E_{1 n} \\ {\bar{E}}_{21} & [1, 1] & . . . & {\bar{E}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\bar{E}}_{n 1} & {\bar{E}}_{n 2} & . . . & (1, 1) \end{matrix})$ (7)

3.2 A method to derive consistent IMPRs

Proposition 1 Let $\bar{A} = {({\bar{a}}_{ij})}_{n \times n}$ be an IMPR. If we have $a_{ij}^{-} a_{ij}^{+} = a_{ik}^{-} a_{ik}^{+} a_{kj}^{-} a_{kj}^{+}$ (8) for all i, j, k = 1, 2, …, n with i < j < k (ori > j > k), then $\bar{A}$ is consistent.

When $\bar{A}$ is consistent, from Formula (8), we know that $a_{ij}^{-} a_{ij}^{+} = \sqrt[n]{\prod_{k = 1}^{n} a_{ik}^{-} a_{ik}^{+} a_{kj}^{-} a_{kj}^{+}}$ (9) for all i, j, = 1, 2, …, nwithi < j.

When $\bar{A}$ is inconsistent, a method is proposed to obtain the consistency matrix with the following steps.

Step 1: Use Formula (9) to calculate the deviation variables denoted by δ_ij, and $δ_{ij} = \frac{\sqrt[n]{\prod_{k = 1}^{n} a_{ik}^{-} a_{ik}^{+} a_{kj}^{-} a_{kj}^{+}}}{a_{ij}^{-} a_{ij}^{+}} \neq 1$ [30].

Step 2: Let λ_ij ∈ [0, 1], then ${\begin{matrix} l_{ij} = a_{ij}^{-} δ_{ij}^{λ_{ij}} \\ u_{ij} = a_{ij}^{+} δ_{ij}^{1 - λ_{ij}} \end{matrix} (i, j, = 1, 2, \dots, n, i < j .)$ (10)

Obviously, in the Formula (10) l_ij = 1/u_ij and u_ij = 1/l_ij. In addition, Formula (10) should satisfy the constraint: l_ij ⩽ u_ij. Namely, $ln a_{ij}^{-} + λ_{ij} ln δ_{ij} ⩽ ln a_{ij}^{+} + (1 - λ_{ij}) ln δ_{ij}$ (11)

Step 3: Since we cannot confirm whether the lower bounds or the upper bounds caused the inconsistency, the adjusted amount should be as equal as possible. Therefore, from the Formula (10), we can derive $a_{ij}^{-} - l_{ij} = a_{ij}^{+} - u_{ij}$ (12) for all i, j, = 1, 2, …, nwithi < j.

Step 4: According to the Equation (11), we derive $λ_{ij} = \frac{ln \frac{1}{2} (\frac{a_{ij}^{-} - a_{ij}^{+}}{a_{ij}^{-}} + \sqrt{{(\frac{a_{ij}^{-} - a_{ij}^{+}}{a_{ij}^{-}})}^{2} + \frac{4 a_{ij}^{+} δ_{ij}}{a_{ij}^{-}}})}{ln δ_{ij}}$ (13) for all i, j, = 1, 2, …, n with i < j.

Because the (11) and (13) may not hold simultaneously in some situation, it is necessary to relax the condition (13). We introduce the deviation variables $η_{ij}^{+}$ and $η_{ij}^{-}$ , $η_{ij}^{+}$ , $η_{ij}^{-} ⩾ 0$ : $λ_{ij} - \frac{ln \frac{1}{2} (\frac{a_{ij}^{-} - a_{ij}^{+}}{a_{ij}^{-}} + \sqrt{{(\frac{a_{ij}^{-} - a_{ij}^{+}}{a_{ij}^{-}})}^{2} + \frac{4 a_{ij}^{+} δ_{ij}}{a_{ij}^{-}}})}{ln δ_{ij}} - η_{ij}^{+} + η_{ij}^{-} = 0$ (14)

Step 5: To ensure the deviation variables $η_{ij}^{+}$ and $η_{ij}^{-}$ are as smaller as possible, we need to construct a goal programming model, as follows: $\min \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (η_{ij}^{+} + η_{ij}^{-})$ $s . t . ln a_{ij}^{-} + λ_{ij} ln δ_{ij} ⩽ ln a_{ij}^{+} + (1 - λ_{ij}) ln δ_{ij}$ $λ_{ij} - \frac{ln \frac{1}{2} (\frac{a_{ij}^{-} - a_{ij}^{+}}{a_{ij}^{-}} + \sqrt{{(\frac{a_{ij}^{-} - a_{ij}^{+}}{a_{ij}^{-}})}^{2} + \frac{4 a_{ij}^{+} δ_{ij}}{a_{ij}^{-}}})}{ln δ_{ij}}$ $- η_{ij}^{+} + η_{ij}^{-} = 0$ $λ_{ij} \in [0, 1], δ_{ij} \neq 1, η_{ij}^{+}, η_{ij}^{-} ⩾ 0, i, j, = 1, 2, \dots, n, i < j$ (15)

Through model (15), the optimal deviation values δ_ij can be derived. Then according to Formula (10), we can obtain the correct values of l_ij and u_ij for all i, j, = 1, 2, …, n with i < j, and further derive the adjusted consistent IMPR ${\bar{A}}^{'}$ , in which ${\bar{A}}^{'} = {({\bar{a}}_{ij}^{'})}_{n \times n} = {([l_{ij}, u_{ij}])}_{n \times n}$ and ${\bar{a}}_{ij}^{'} = 1 / {\bar{a}}_{ji}^{'}$ . After having the adjusted consistent IMPR ${\bar{A}}^{'}$ , we can rank DMUs.

In addition to the above method, there are other ways to get the appropriate adjusted values, for example, the extreme method in which λ_ij = 1 or λ_ij = 0; the mean method where λ_ij = 0.5; the proportional method where $λ_{ij} = \frac{l_{ij}}{l_{ij} + u_{ij}}$ . In practice, we can choose different methods according to the situation. Either way, it is strictly necessary to make sure the constraint (11) is valid.

Then the consistent IMPR can be denote by ${\bar{E}}^{'} = {({\bar{E}}_{ij}^{'})}_{n \times n} = {([E_{ij}^{' -}, E_{ij}^{' +}])}_{n \times n}$ . For each ${\bar{E}}_{ij}^{'}$ , the interval priority weights ${\bar{w}}_{d}$ can be calculated by ${\bar{w}}_{d} = [w_{d}^{l}, w_{d}^{u}]$ $= [\frac{\sum_{j = 1}^{n} E_{dj}^{' -}}{\sum_{j = 1}^{n} E_{dj}^{' -} + \sum_{k = 1, k \neq d}^{n} \sum_{j = 1}^{n} E_{kj}^{' +}},$ $\frac{\sum_{j = 1}^{n} E_{dj}^{' +}}{\sum_{j = 1}^{n} E_{dj}^{' +} + \sum_{k = 1, k \neq d}^{n} \sum_{j = 1}^{n} E_{kj}^{' -}}]$ (16)

Next, we use the compared Formula (17) for intervals in Meng et al. [34] to calculate the preferred degree $p_{ij} = p ({\bar{w}}_{i} > {\bar{w}}_{j})$ , $p ({\bar{w}}_{i} > {\bar{w}}_{j}) = {\begin{matrix} 1 w_{i}^{l} ⩾ w_{j}^{u} \\ 1 - \frac{(w_{j}^{u} - w_{i}^{l})^{2}}{2 d ({\bar{w}}_{i}) d ({\bar{w}}_{j})} w_{j}^{l} ⩽ w_{i}^{l} ⩽ w_{j}^{u} ⩽ w_{i}^{u} \\ \frac{2 w_{i}^{u} - (w_{j}^{u} + w_{j}^{l})}{2 d ({\bar{w}}_{i})} w_{i}^{l} ⩽ w_{j}^{l} ⩽ w_{j}^{u} ⩽ w_{i}^{u} \end{matrix}$ (17) for all i, j, = 1, 2, …, nwithi < j, and $d ({\bar{w}}_{i}) = w_{i}^{u} - w_{j}^{l}$ , $d ({\bar{w}}_{j}) = w_{j}^{u} - w_{j}^{l}$ .

Last, we can construct the preferred degree matrix P = (p_ij) _n×n according to the preferred degree p_ij, then DMUs can be ranked by $r_{d} = \sum_{j = 1}^{n} p_{dj}, d = 1, 2, \dots, n$ , which is the sum of each line of p.

4 Empirical illustrations

With the rapid development of the economy, the hotel industry faced fierce competition among peers, so it is time to improve the core competitiveness of hotels. Operating efficiency is one of the best choices on assessing hotel operating conditions, how to strengthen the hotel’s internal management, optimize the allocation of resources and achieve the ideal of input and output efficiency to enhance their sustainable competitive ability becomes important issues that managers and decision-makers are facing. The performance of hotels has been paid much attention by enterprises, government departments, and scholars.

The existing research literature shows that data envelopment analysis (DEA) can be suitably used to measure the efficiency of the hotel. The use of the DEA model to evaluate hotel operating efficiency began in the 1990s in foreign countries. Anderson et al. [35] first used BBC and CCR models in DEA theory to study the technical efficiency, pure technical efficiency, and allocative efficiency of 48 American hotels. Hwang and Chang [36] firstly combined the DEA method and Malmquist index method to conduct dynamic analysis on the management efficiency of 45 hotels in Taiwan from 1994 to 1998, which expanded the theory of data envelopment analysis method. Wang et al. [37] used the cost model in DEA to measure the performance of 49 hotels in Taiwan. Wu et al. [38] combined with the non-radial DEA model, evaluated the performance of the operation efficiency of 23 tourist hotels in Taipei in 2006, and put forward the methods to improve the performance. In addition, there are other studies that apply the DEA method to hotel performance evaluation (see Tsaur [39], Chiang et al. [40], Chiang [41]).

As discussed above, DEA has become an important technique for analyzing hotel performance. Therefore, this paper takes a sample of 12 four- or five-plum tourist hotels in Taipei and uses the method proposed in this paper to analyze performance. The original data is derived from Taiwan Tourism Bureau. In this article, we use two input variables and three output variables to evaluate the tourist hotels in the sample. Input variables include the total number of employees (x₁), total number of guest rooms (x₂), and output variables include room revenues (y₁), food and beverage (F&B) revenues (y₂), and other revenues (y₃). In the output variables, the contribution of the room revenues department and food and beverage (F&B) department to the total income is quite large. According to the data of the Taiwan Tourism Administration, the average contribution rates of the room revenues and the food and beverage (F&B) are 46.54% and 40.54% respectively, which indicates that our selection of indicators is reasonable. In this application, a DMU refers to an international tourist hotel (ITH). Table 1 provides a list of the hotels in our sample and Table 2 shows the data of input and output variables.

Table 1
List of sample hotels

DMU Hotel abbreviation Hotel

1 GRA The Grand Hotel

2 IMP Imperial Hotel Taipei

3 GLP Gloria Prince Hotel

4 EMP Emperor Hotel

5 RIV Riverview Taipei

6 CAP Caesar Park Taipei

7 GOC Golden China Hotel

8 SAW San Want Hotel

9 BRO Brother Hotel

10 SAN Santos Hotel

11 LAN The Landis Taipei Hotel

12 UNI United Hotel

DMU	Hotel abbreviation	Hotel
1	GRA	The Grand Hotel
2	IMP	Imperial Hotel Taipei
3	GLP	Gloria Prince Hotel
4	EMP	Emperor Hotel
5	RIV	Riverview Taipei
6	CAP	Caesar Park Taipei
7	GOC	Golden China Hotel
8	SAW	San Want Hotel
9	BRO	Brother Hotel
10	SAN	Santos Hotel
11	LAN	The Landis Taipei Hotel
12	UNI	United Hotel

Table 2

Data of ITHs

DMU	Hotel abbreviation	x₁	x₂	y₁	y₂	y₃
1	GRA	3,000	3,653	284,869,544	340,804,635	75,346,464
2	IMP	1,956	1,269	135,834,810	68,471,923	13,899,670
3	GLP	1,258	1,428	75,797,472	69,018,560	18,819,451
4	EMP	582	402	19,039,899	5,096,106	470,285
5	RIV	1,254	404	57,859,959	8,954,977	1,752,976
6	CAP	2,868	1,693	271,883,550	130,790,602	27,188,097
7	GOC	1,290	1,029	61,563,295	69,070,454	4,987,191
8	SAW	1,608	2,134	104,926,477	182,841,485	24,291,526
9	BRO	1500	2,940	106,527,541	262,362,237	28,429,418
10	SAN	1631	887	60,078,574	28,314,655	12,415,067
11	LAN	1,254	1,514	107,408,796	71,679,912	29,892,324
12	UNI	1458	740	62,372,840	8,265,426	7,441,327

Note: x₁ = total number of employees; x₂ = total number of guest rooms; y₁ = room revenues (in NT$); y₂ = F&B revenues (in NT$); and y₃ = other revenues (in NT$).

Firstly, we use the CCR model and SBM model to calculate the efficiency of the case in this paper, so as to compare with the measurement results of our model. The third and fourth columns of Table 3 are the calculated results of the CCR model and SBM model respectively. And the efficiency interval in the last column of Table 3 is derived from the method in this article. Although the efficiency values calculated by each model are different, the three models are highly consistent in judging the effectiveness of the DMUs. According to Table 3, three hotels are evaluated as efficient among the 12 tourist hotels in all the models. Both the upper and lower efficiency values of the efficiency interval on the efficient DMU are 1 in the three hotels, namely, they are totally efficient. In addition, the evaluation results of the three models also have slight differences. The efficiency value of the CCR model is larger than that of the SBM model, this is because the traditional CCR model always maximizes the efficiency value. In other words, the efficiency value of the CCR model is derived from the optimistic viewpoint [42]. Compared with our model, the efficiency value obtained by the CCR model is within the range of efficiency interval or greater than the upper limit of the efficiency interval.

Table 3

Efficiency scores of DMUs

DMU	Hotel abbreviation	CCR efficiency	SBM efficiency	Efficiency interval
1	GRA	1	1	[1, 1]
2	IMP	0.7324	0.5957	[0.5957, 0.7159]
3	GLP	0.6462	0.5838	[0.5838, 0.6694]
4	EMP	0.345	0.0883	[0.0883, 0.1402]
5	RIV	0.8918	0.2605	[0.2605, 0.2791]
6	CAP	1	1	[1, 1]
7	GOC	0.7261	0.3573	[0.3573, 0.4649]
8	SAW	0.9272	0.6991	[0.6991, 0.8451]
9	BRO	1	1	[1, 1]
10	SAN	0.7114	0.3912	[0.3912, 0.4853]
11	LAN	0.9572	0.726	[0.726, 0.7375]
12	UNI	0.5901	0.1933	[0.1933, 0.2678]

Based on the theoretical analysis in section 2 and section 3, we can use the following steps to carry out the full ordering of these DMUs.

Step 1: Based on the model (1) and model (3), we can obtain the upper and lower bounds of efficiency values of DMU_d (d = 1, …, n) respectively. The efficiency interval of DMUs is given in the last column of Table 3.

Step 2: After the interval efficiency is obtained, the IMPR $\bar{E}$ can be calculated by Equation (5).

Step 3: Using the formula (8), we can check that the IMPR $\bar{E}$ is consistent. The consistent IMPR ${\bar{E}}^{'}$ is shown in Table 4.

Table 4

The consistent IMPR ${\bar{E}}^{'}$

DMU	1	2	3	4	5	6	7	8	9	10	11	12
1	[1, 1]	[1.397, 1.679]	[1.494, 1.713]	[7.133, 11.325]	[3.583, 3.839]	[1, 1]	[2.151, 2.799]	[1.183, 1.43]	[1, 1]	[2.061, 2.556]	[1.356, 1.377]	[3.734, 5.173]
2	[0.596, 0.716]	[0.832, 1.202]	[0.89, 1.226]	[4.249, 8.108]	[2.134, 2.748]	[0.596, 0.716]	[1.281, 2.004]	[0.705, 1.024]	[0.596, 0.716]	[1.227, 1.83]	[0.808, 0.986]	[2.224, 3.704]
3	[0.584, 0.669]	[0.815, 1.124]	[0.872, 1.147]	[4.164, 7.581]	[2.092, 2.57]	[0.584, 0.669]	[1.256, 1.873]	[0.691, 0.958]	[0.584, 0.669]	[1.203, 1.711]	[0.792, 0.922]	[2.18, 3.463]
4	[0.088, 0.14]	[0.123, 0.235]	[0.132, 0.24]	[0.63, 1.588]	[0.316, 0.538]	[0.088, 0.14]	[0.19, 0.392]	[0.104, 0.201]	[0.088, 0.14]	[0.182, 0.358]	[0.12, 0.193]	[0.33, 0.725]
5	[0.261, 0.279]	[0.364, 0.469]	[0.389, 0.478]	[1.858, 3.161]	[0.933, 1.071]	[0.261, 0.279]	[0.56, 0.781]	[0.308, 0.399]	[0.261, 0.279]	[0.537, 0.713]	[0.353, 0.384]	[0.973, 1.444]
6	[1, 1]	[1.397, 1.679]	[1.494, 1.713]	[7.133, 11.325]	[3.583, 3.839]	[1, 1]	[2.151, 2.799]	[1.183, 1.43]	[1, 1]	[2.061, 2.556]	[1.356, 1.377]	[3.734, 5.173]
7	[0.357, 0.465]	[0.499, 0.78]	[0.534, 0.796]	[2.549, 5.265]	[1.28, 1.785]	[0.357, 0.465]	[0.769, 1.301]	[0.423, 0.665]	[0.357, 0.465]	[0.736, 1.188]	[0.484, 0.64]	[1.334, 2.405]
8	[0.699, 0.845]	[0.977, 1.419]	[1.044, 1.448]	[4.986, 9.571]	[2.505, 3.244]	[0.699, 0.845]	[1.504, 2.365]	[0.827, 1.209]	[0.699, 0.845]	[1.441, 2.16]	[0.948, 1.164]	[2.611, 4.372]
9	[1, 1]	[1.397, 1.679]	[1.494, 1.713]	[7.133, 11.325]	[3.583, 3.839]	[1, 1]	[2.151, 2.799]	[1.183, 1.43]	[1, 1]	[2.061, 2.556]	[1.356, 1.377]	[3.734, 5.173]
10	[0.391, 0.485]	[0.546, 0.815]	[0.584, 0.831]	[2.79, 5.496]	[1.402, 1.863]	[0.391, 0.485]	[0.841, 1.358]	[0.463, 0.694]	[0.391, 0.485]	[0.806, 1.241]	[0.53, 0.668]	[1.461, 2.511]
11	[0.726, 0.738]	[1.014, 1.238]	[1.085, 1.263]	[5.178, 8.352]	[2.601, 2.831]	[0.726, 0.738]	[1.562, 2.064]	[0.859, 1.055]	[0.726, 0.738]	[1.496, 1.885]	[0.984, 1.016]	[2.711, 3.815]
12	[0.193, 0.268]	[0.27, 0.45]	[0.289, 0.459]	[1.379, 3.033]	[0.693, 1.028]	[0.193, 0.268]	[0.416, 0.75]	[0.229, 0.383]	[0.193, 0.268]	[0.398, 0.685]	[0.262, 0.369]	[0.722, 1.385]

Step 4: Combined with the consistency IMPR ${\bar{E}}^{'}$ , the priority weights can be obtained by using formula (16), as shown in Table 5.

Table 5

The priority weights ${\bar{w}}_{d}$

${\bar{w}}_{1}$	${\bar{w}}_{2}$	${\bar{w}}_{3}$	${\bar{w}}_{4}$	${\bar{w}}_{5}$
[0.1052, 0.1793]	[0.0629, 0.1277]	[0.0613, 0.1202]	[0.0091, 0.0258]	[0.0269, 0.0514]
${\bar{w}}_{6}$	${\bar{w}}_{7}$	${\bar{w}}_{8}$	${\bar{w}}_{9}$	${\bar{w}}_{10}$
[0.1052, 0.1793]	[0.0374, 0.0839]	[0.0743, 0.1494]	[0.1052, 0.1793]	[0.0409, 0.0877]
${\bar{w}}_{11}$	${\bar{w}}_{12}$
[0.0759, 0.1334]	[0.02, 0.0489]

Step 5: Based on the priority weights we got in the previous step, the preferred degree matrix P = (p_ij) _n×n is derived, as shown in Table 6.

Table 6

The preferred degree matrix P

DMU	1	2	3	4	5	6	7	8	9	10	11	12
1	0.5	0.9473	0.9742	1	1	0.5	1	0.8242	0.5	1	0.9066	1
2	0.0527	0.5	0.5696	1	1	0.0527	0.9269	0.2926	0.0527	0.8988	0.3602	1
3	0.0258	0.4304	0.5	1	1	0.0258	0.907	0.238	0.0258	0.874	0.2901	1
4	0	0	0	0.5	0	0	0	0	0	0	0	0.0347
5	0	0	0	1	0.5	0	0.0859	0	0	0.0479	0	0.6562
6	0.5	0.9473	0.9742	1	1	0.5	1	0.8242	0.5	1	0.9066	1
7	0	0.0731	0.093	1	0.9141	0	0.5	0.0131	0	0.4249	0.0121	0.9504
8	0.1758	0.7074	0.762	1	1	0.1758	0.9869	0.5	0.1758	0.9746	0.5963	1
9	0.5	0.9473	0.9742	1	1	0.5	1	0.8242	0.5	1	0.9066	1
10	0	0.1012	0.126	1	0.9521	0	0.5751	0.0254	0	0.5	0.026	0.9761
11	0.0934	0.6398	0.7099	1	1	0.0934	0.9879	0.4037	0.0934	0.974	0.5	1
12	0	0	0	0.9653	0.3438	0	0.0496	0	0	0.0239	0	0.5

Step 6: By adding up each row in the preferred degree matrix P, we can obtain the total preferred degree of each DMU.

According to the preferred degree matrix P in Table 6 and the total preferred degree in Fig. 1, the full order of the DMUs is shown as follows:

${DMU}_{1} ≻_{0.5} {DMU}_{6} ≻_{0.5} {DMU}_{9} ≻_{0.8242} {DMU}_{8} ≻_{0.5963} {DMU}_{11} ≻_{0.6398} {DMU}_{2} ≻_{0.5696}$ ${DMU}_{3} ≻_{0.874} {DMU}_{10} ≻_{0.5751} {DMU}_{7} ≻_{0.9141} {DMU}_{5} ≻_{0.6562} {DMU}_{12} ≻_{0.9653} {DMU}_{4}$

As is shown above, there are three DMUs that are totally efficient with the preferred degree of 0.5 among each other. This means that all the three DMUs perform best, and there is no significant difference between them. In other words, DMU₁, DMU₆, and DMU₉ are all the projection points on the Pareto-efficient frontier, which can serve as benchmarks for the improvements of other DMU. The preferred degree measures the degree of superiority between two adjacent DMUs. The greater the preferred degree between two adjacent DMU, the better the former is than the latter. Taking DMU₉ and DMU₈ for example, as shown above, the preference degree of DMU₉ over DMU₈ is 0.8242, this represents that the DMU₉ is significantly better than DMU₈. Note that the preference degree among theDMU₁₂ and DMU₄ is the largest, which means DMU₁₂ and DMU₄ have the biggest gap.

Fig. 1

The total preferred degree of each DMU.

According to the complete ordering of DMUs, we can compare the operating conditions of 12 hotels. Among the 12 hotels, The Grand Hotel, Caesar Park Taipei, and Brother Hotel have the best performance, with an efficiency interval of [1, 1]. These three performed slightly better than San Want Hotel with the preferred degree of 0.8242, which means there is a significant gap between this hotel and the three hotels with the best efficiency. We can see that San Want Hotel, The Landis Taipei Hotel, Imperial Hotel Taipei, and Gloria Prince Hotel rank fourth, fifth, sixth, and seventh, and the efficiency gaps between them are not big because they all have the preference below 0.65. In addition, the performance of Santos Hotel is obviously inferior to Gloria Prince Hotel, and the performance of Riverview Taipei in 10th place is obviously inferior to Golden China Hotel in 9th place. Among the 12 hotels, the worst performer is the Emperor Hotel, which ranks last. And Emperor Hotel has the biggest gap with United Hotel in the eleventh place, with the preferreddegree 0.9653.

This ranking order can aid the CEO to find the best-performed hotels, i.e., The Grand Hotel, Caesar Park Taipei, and Brother Hotel as a benchmark, other hotels can learn from them to improve their performance. Through the above analysis, we can find that Emperor Hotel has the worst performance, while at the same time has a huge room for improvement, so the operation and management methods need to be rectified and improved so as to achieve rapid development. Next, there are still have some poor performing hotels, such as Riverview Taipei, United Hotel, and Golden China Hotel, they also need to pay close attention to their operations and take some active action quickly. Then some hotels perform better, i.e., San Want Hotel and The Landis Taipei Hotel, and they can learn from the best performing hotels to get better results.

Using the method introduced in this paper, we not only calculate the optimal efficiency and the worst efficiency of 12 hotels, but also make a full ranking of 12 hotels according to the efficiency interval combined with the AHP method. The poor ranking of some hotels indicates low efficiency. In order to improve efficiency, hotels can take some valid measures, such as improving the production technology, enhancing the management level, changing the management mode, or strengthening the training of employees to improve the efficiency of existing resources. In particular, reducing input and increasing output can effectively improve efficiency when DEA is used for performance evaluation. But we also need to combine the specific situation when evaluating hotel performance. For smaller hotels, we can focus on the output, and it is a better choice to increase output while keeping input unchanged. For example, Emperor Hotel has the lowest ranking, but its investment is the lowest compared to other hotels, with only 582 employees and 402 guest rooms. Obviously, the best way to increase efficiency is to increase output such as increasing room occupancy. For large-scale hotels, we think it is appropriate to reduce the input to avoid unnecessary waste of resources. For example, United Hotel and Santos Hotel have more than 1,400 employees and 700 rooms while doing poorly in the rankings, so they can consider cutting back to improve their performance.

5 Conclusions

The interval efficiency method solves the uncertainty of efficiency evaluation. While the previous research deal with the uncertainty in efficiency measurement by applying radial interval efficiency model, few of them employ non-radial interval efficiency model to describe the uncertainty in efficiency measurement. In the non-radial measurement, the SBM model proposed by Tone (2001) is widely employed because of its desirable properties, such as unit invariant, strong monotonic and reference-set dependent. However, it also suffers from one drawback, that is, it only considers the furthest target from the strongly efficient frontier. Another issue in the study of interval DEA is the performance ranking of DMUs. The interval DEA is not able to full rank DMUs because the interval efficiency may be incomparable in some cases. To remedy the above problems, this paper proposes an interval efficiency method with slacks-based measure, under which the furthest and closest targets are considered. After obtaining the interval efficiency, an AHP/DEA method is proposed to process the performance ranking of DMUs.

Compared with previous methods, this paper has the following properties. Firstly, we obtained the efficiency interval of the non-radial model. The upper bound of the interval is calculated by the SBM model, and the lower bound of the interval is derived from the closest targets model. Therefore, the efficiency interval obtained from our method sufficiently takes into account the uncertainty of the decision-maker compared with the previous SBM model. Secondly, AHP is used to realize the full ranking of DMUs. The interval multiplicative preference relations matrix constructed in this paper considers all the possible efficiency pairwise comparisons of each pair of DMUs. Last but not least, the preference relationship is calculated by the DEA model and it eliminates the subjectivity of pairwise comparison between DMUs in AHP.

There are some noteworthy directions for future works. Firstly, we can consider how to develop our approach to apply in both crisp and fuzzy environments. Secondly, the practical application of this method can be extended to other fields, such as supplier selection in supply chain management, supplier selection in outsourcing, and inventory selection in finance.

Footnotes

Acknowledgments

The research is supported by National Natural Science Foundation of China (No.71871223), Innovation - Driven Planning Foundation of Central South University (2019CX041).

References

Charnes

, Cooper

W.W.

and Rhodes

, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978), 429–444.

Banker

R.D.

, Charnes

and Cooper

W.W.

, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science 30 (1984), 1078–1092.

Seiford

L.M.

and Zhu

, Infeasibility of super-efficiency data envelopment analysis models, INFOR: Information Systems and Operational Research 37 (1999), 174–187.

Lozano

and Villa

, Data envelopment analysis of integer-valued inputs and outputs, Computers & Operations Research 33 (2006), 3004–3014.

Charnes , Cooper

W.W.

, Golany

and Seiford

, et al., Foundations of Data Envelopment Analysis for Pareto-Koopmans Efficient Empirical Productions Functions, Texas Univ at Austin Center for Cybernetic Studies, 1985.

Hashimoto, A ranked voting system using a DEA/AR exclusion model: A note, European Journal of Operational Research 97 (1997), 600–604.

Doyle

and Green

, Efficiency and cross-efficiency in DEA: Derivations, meanings and uses, Journal of the Operational Research Society 45 (1994), 567–578.

Ghasemi

, Najafi

, Lotfi

F.H.

and Sobhani

F.M.

, Assessing the performance of organizations with the hierarchical structure using data envelopment analysis: An efficiency analysis of Farhangian University, Measurement 156 (2020), 107609.

Fukuyama

and Matousek

, Nerlovian revenue inefficiency in a bank production context: evidence from Shinkin banks, European Journal of Operational Research 271 (2018), 317–330.

10.

Zhou

, Xu

, Chai

, Yao

, et al., Efficiency evaluation for banking systems under uncertainty: A multi-period three-stage DEA model, Omega 85 (2019), 68–82.

11.

Jiang

, Zhang

and Jin

, Sustainability efficiency assessment of listed companies in China: a super-efficiency SBM-DEA model considering undesirable output, Environmental Science and Pollution Research (2021), 1–17.

12.

Wang

and Wang

, Interval enhanced Russell measure with undesirable outputs based on data envelopment analysis: An efficiency measurement of industry in China, Journal of Intelligent & Fuzzy Systems (2021), 1–13.

13.

Badiezadeh

, Saen

R.F.

and Samavati

, Assessing sustainability of supply chains by double frontier network DEA: A big data approach, Computers & Operations Research 98 (2018), 284–290.

14.

Omrani

, Amini

and Alizadeh

, An integrated group best-worst method–Data envelopment analysis approach for evaluating road safety: A case of Iran, Measurement 152 (2020), 107330.

15.

Entani

, Maeda

and Tanaka

, Dual models of interval DEA and its extension to interval data, European Journal of Operational Research 136 (2002), 32–45.

16.

Amirteimoori , Kordrostami

and Rezaitabar

, An improvement to the cost efficiency interval: A DEA-based approach, Applied Mathematics and Computation 181 (2006), 775–781.

17.

Azizi

, The interval efficiency based on the optimistic and pessimistic points of view, Applied Mathematical Modelling 35 (2011), 2384–2393.

18.

Azizi

, DEA efficiency analysis: A DEA approach with double frontiers, International Journal of Systems Science 45 (2013), 2289–2300.

19.

Doyle

and Green

, Efficiency and cross-efficiency in DEA: Derivations, meanings and uses, Journal of the Operational Research Society 45 (1994), 567–578.

20.

Liu

S.T.

, A DEA ranking method based on cross-efficiency intervals and signal-to-noise ratio, Annals of Operations Research 261 (2018), 207–232.

21.

, Meng

and Xiong

, Interval cross efficiency for fully ranking decision making units using DEA/AHP approach, Annals of Operations Research 271 (2018), 297–317.

22.

Yang

, Ang

, Xia

and Yang

, Ranking DMUs by using interval DEA cross efficiency matrix with acceptability analysis, European Journal of Operational Research 223 (2012), 483–488.

23.

Ang

, An

, Yang

and Yang

, Ranking ranges, performance maverick and diversity for decision-making units with interval cross-efficiency matrix, International Journal of Systems Science: Operations & Logistics 7 (2020), 46–59.

24.

Tone

, A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research 130 (2001), 498–509.

25.

Mirhedayatian

and Saen

R.F.

, A new approach for weight derivation using data envelopment analysis in the analytic hierarchy process, Journal of the Operational Research Society 62 (2011), 1585–1595.

26.

Dyer

J.S.

, Remarks on the analytic hierarchy process, Management Science 36 (1990), 249–258.

27.

Winkler

R.L.

, Decision modeling and rational choice: AHP and utility theory, Management Science 36 (1990), 247–248.

28.

Sinuany-Stern

, Mehrez

and Hadad

, An AHP/DEA methodology for ranking decision making units, International Transactions in Operational Research 7 (2000), 109–124.

29.

Aparicio

, Ruiz

J.L.

and Sirvent

, Closest targets and minimum distance to the Pareto-efficient frontier in DEA, Journal of Productivity Analysis 28 (2007), 209–218.

30.

Jin

, Ni

, Langari

and Chen

, Consistency Improvement-Driven Decision-Making Methods with Probabilistic Multiplicative Preference Relations, Group Decision & Negotiation 29 (2020).

31.

Jin

, Liu

, Zhou

and Martínez

, Consensus-based linguistic distribution large-scale group decision making using statistical inference and regret theory, Group Decision and Negotiation (2021), 1–33.

32.

Saaty

T.L.

and Vargas

L.G.

, Uncertainty and rank order in the analytic hierarchy process, European Journal of Operational Research 32 (1987), 107–117.

33.

Saaty

, The analytic hierarchy process (AHP) for decision making. in Kobe, Japan, 1980.

34.

Meng

, Zeng

and Li

, Research the priority methods of interval numbers complementary judgment matrix. in 2007 IEEE International Conference on Grey Systems and Intelligent Services. 2007. IEEE.

35.

Anderson

R.I.

, Fish

, Xia

and Michello

, Measuring efficiency in the hotel industry: A stochastic frontier approach, International Journal of Hospitality Management 18 (1999), 45–57.

36.

Hwang

S.N.

and Chang

T.Y.

, Using data envelopment analysis to measure hotel managerial efficiency change in Taiwan, Tourism Management 24 (2003), 357–369.

37.

Wang

F.C.

, Hung

W.T.

and Shang

J.K.

, Measuring the cost efficiency of international tourist hotels in Taiwan, Tourism Economics 12 (2006), 65–85.

38.

, Tsai

and Zhou

, Improving efficiency in international tourist hotels in Taipei using a non-radial DEA model, International Journal of Contemporary Hospitality Management (2011).

39.

Tsaur

S.H.

, The operating efficiency of international tourist hotels in Taiwan, Asia Pacific Journal of Tourism Research 6 (2001), 73–81.

40.

Chiang

, Tsai

and Wang

L.S.-M.

, A DEA evaluation of Taipei hotels, Annals of Tourism Research 31 (2004), 712–715.

41.

Chiang

, A hotel performance evaluation of Taipei international tourist hotels–using data envelopment analysis, Asia Pacific Journal of Tourism Research 11 (2006), 29–42.

42.

Entani

and Tanaka

, Improvement of efficiency intervals based on DEA by adjusting inputs and outputs, European Journal of Operational Research 172 (2006), 1004–1017.