A novel slacks-based interval DEA model and application

This paper proposes a novel slacks-based interval DEA approach that computes interval targets, slacks, and crisp inefficiency scores. It uses interval arithmetic and requires solving a mixed-integer linear program. The corresponding super-efficiency formulation to discriminate among the efficient units is also presented. We also provide a case study of its application to sustainable tourism in the Mediterranean region, assessing the sustainable tourism efficiency of twelve Mediterranean regions to validate the proposed approach. The inputs and outputs cover the three sustainability dimensions and include GHG emissions as an undesirable output. Three regions were found inefficient, and the corresponding inputs and output improvements were computed. A total rank of the regions was also obtained using the super-efficiency model.


Introduction
Data Envelopment Analysis (DEA) is a well-known non-parametric methodology to evaluate the efficiency of a set of Decision-Making Units (DMUs) that consume inputs (i.e., resources) to produce outputs (Zhu [55], Cooper et al. [14]).A Production Possibility Set (PPS) is derived from the observed inputs and outputs using certain axioms and the Principle of Minimum Extrapolation.
This PPS contains all the feasible operating points.The non-dominated subset of the PPS is called the efficient frontier.A DMU is efficient if it lies on the efficient frontier.Otherwise, it is inefficient and can be projected onto a target operating point on the efficient frontier.
Regarding super-efficiency, aimed at discriminating between efficient DMUs, it was introduced by Andersen and Petersen [2] and has been continued by many authors such as Zhu [54], Seiford and Zhu [45], Zhong et al. [53], Li et al. [37], Esteve et al. [16], Bolos et al. [9], among others.In comparison, the initial super-efficiency approaches involved radial DEA models, a super-slacks-based measure of efficiency (super-SBM, [49]), and a super Slack-based measure of inefficiency (super-SBI, [41]) have also been proposed.This paper considers inputs and outputs with continuous interval-valued data as mathematical uncertainty modeling.The most similar and recent DEA model is that by Arana-Jimenez et al. [4].In that work, authors consider a hybrid scenario in which, in addition to integer interval data, some inputs or outputs are given as continuous intervals, but no super-efficiency model is proposed to rank or discriminate the efficient DMUs.In the present work, we focus on continuous cases, with an enhanced formulation of the model given in [4] and a super-efficiency model.Thus, while [4] uses two phases for the slacks-based model, under an additive and non-oriented approach, we propose a one-phase interval slack-based model in the present work.This is possible by using the gh-difference of two intervals.Also, we propose a super SBI interval model to discriminate between efficient units.
To illustrate the usefulness of the proposed approach, we present an application to sustainable tourism.The first definitions of sustainable tourism focused on environmental and economic development, with community involvement included later [21].The current sustainable tourism policy is often economic-growth oriented, with theoretical differences from sustainable development [19].However, sustainable tourism management policies should maximize economic benefits while minimizing adverse environmental impacts [42].To the latter, GHG emissions are considered undesirable output, discussed in the application section and included in our model.To this matter, Tone, and Tsutsui [50] propose a model in which undesirable output variables are treated as inputs, as well as the cases of [40] and [12], considering water consumption and tourism energy, respectively.Finally, tourism sustainability should not only be based on a guide of good practices; it must also consider quantitative data that allows evaluation with which to make decisions.In this regard, using indicators is a crucial tool that enables an assessment of the transition toward Sustainability [1].
The structure of the paper is the following.Section 2 introduces the con-ventional crisp production possibility set (PPS) and a slacks-based DEA model for efficiency assessment.Section 3 introduces continuous intervals, especially arithmetic operations, and partial orders.Section 4 presents the continuous interval PPS and Slack-based measures of inefficiency.In Section 5, a new Enhanced interval slacks-based DEA approach is proposed.Section 6 presents the corresponding Super SBI interval DEA model to discriminate among efficient units.Section 7 presents the application to sustainable tourism, and finally, Section 8 summarizes and concludes.

Crisp production possibility set and Slack-based measure
Let us consider a set of n DMUs.For j P J " t1, . . ., nu, each DMU j has m inputs X j " px 1 j , . . ., x mj q P R m , produces s outputs Y j " py 1 j , . . ., y s j q P R s .In the classic Charnes et al. [11] DEA model, the production possibility set (PPS) or technology, denoted by T, satisfies the following axioms: (A1) Envelopment: pX j , Y j q P T, for all j P J.
Following the minimum extrapolation principle (see [7]), the DEA PPS, which contains all the feasible input-output bundles, is the intersection of all the sets that satisfy axioms (A1)-(A4) or axioms (A1)-(A3) can be expressed,respectively, as - -. In λ j y r j ě y rp `sy r , r " 1, . . ., S, where λ j , j " 1, . . ., n, are the intensity variables used for defining the corresponding efficient target of DMU p and the constraint ř N j"1 λ j " 1 only applies to the VRS case, not to the CRS case.The inefficiency measures IpX p , Y p q is units invariant and non-negative.Moreover, a DMU p is efficient if and only if IpX p , Y p q " 0.

Notation and preliminaries
This paper presents uncertainty on the production possibility set by modeling the corresponding inequality relationships using partial orders on integer intervals.This requires introducing first the following notation and results.
Let R be the real number set.We denote by K C " " a, a ‰ | a, a P R and a ď a ( the family of all bounded closed intervals in R. Some useful and necessary arithmetic operations for the purpose of this manuscript are described following (see, for instance, [46,47]).
Definition 1.Let A " ra, as P K C , and B " rb, bs P K C • Addition: A `B :" ta `b | a P A, b P Bu " ra `b, a `bs, • Opposite value: ´A " t´a : a P Au " r´a, ´as, • Substraction: A ´B :" ta ´b | a P A, b P Bu " ra ´b, a ´bs, • Multiplication: A ¨B :" ta ¨b | a P A, b P Bu " rminta ¨b, a ¨b, a ¨b, a bu , maxta ¨b, a ¨b, a ¨b, a ¨bus.
Note that A ´A 0, in general.To overcome this issue, when the modeling requires it, we have the gH-difference of two intervals A and B, which we recall from [46,47], as follows: Note that the difference between an interval and itself is zero, that is, A a gH A " r0, 0s.Furthermore, the gH-difference of two intervals always exists and is equal to A a gH B " rminta ´b, a ´bu, maxta ´b, a ´bus Ă A ´B. (3) Later, in the next section, we discuss the modeling with slack variables, and we will recover the difference between intervals to that matter.It is also necessary to define a partial order relationship for integer intervals.To this aim, we will adapt LU-fuzzy partial orders on intervals, which are well-known in the literature, (see, e.g., [52,46] and the references therein).
Definition 2. Given two intervals A " ra, as, B " rb, bs P K C , we say that: Similarly, we define the relationships A B and A ą B for intervals, meaning B A and B ă A, respectively.We denote K C as the set of all non-negative intervals, that is, K C " tA P K C : A 0u.

Proposed interval PPS and Slack-based measure of inefficiency
Let us consider a set of N DMUs, DMU j for j P t1, . . ., Nu, with M inputs X P pK C q M , with x i j " rx i j , x i j s P K C for all i P O X " t1, . . ., Mu, and S outputs Y P pK C q S , with y r j " ry r j , y r j s P K C for all r P O Y " t1, . . ., Su.And the following axioms, are analogous to (A1)-(A4) in Section 2 but consider interval inputs and outputs.We apply the corresponding interval arithmetic, Definition 1, and partial order introduced in Definition 2: (B1) Envelopment: pX j , Y j q P T, for all j P J.
In the case of convexity, but not scalability axiom, the PPS is as follows.
Theorem 2. Under axioms (B1), (B2), and (B3), the VRS interval production possibility set PPS that results from the minimum extrapolation principle is -Proof.This is similar to the proof given by Arana-Jimenez et al. [4].
Definition 3. A DMU p is said to be efficient if and only if for any px, yq P T IDEA , with x X p and y Y p implies px, yq " pX p , Y p q.
Given the T IDEA , we can recover the following slacks-based measure of the inefficiency interval DEA (IDEA) model with two phases.
If a DMU p is efficient, then IpX p , Y p q " 0 (see Arana-Jimenez et al.

[4]) but
IpX p , Y p q " 0 is insufficient to guarantee a DMU's efficiency.Therefore, given the optimal solution of (4), ps x ˚, s y ˚, λ ˚q, we proceed with the phase II of the method to exhaust all remaining input and output slacks.
Given a DMU p with IpX p , Y p q " 0, then HpX p , Y p q " 0 if and only if DMU p is efficient.In other words, a DMU p is efficient if and only if both IpX p , Y p q " 0 and HpX p , Y p q " 0. Let ps x ˚, s y ˚, λ ˚q be the optimal solution of (4), and , or phase 2, was necessary to guarantee the efficiency of the corresponding DMU, as well a to provide their targets.This is because the feasible slacks improvements of the inequality constraints in (4) are not exhausted.Given two positive intervals A, B P K C , with A B, the positive interval upper slack su P K C such that A " B ´su, or the lower slack sl P K C such that A `sl " B do not necessarily exist.For instance, if we consider A " r2, 3s and B " r2, 5s, then A " r2, 3s " B ´su " r2 ´su, 5 ´sus.It is not possible to find a positive interval upper slack su holding the equality.But from A`sl " r2 `sl, 3 `sls " r2, 5s " B, we can find sl " r0, 2s.Similarly, for A " r2, 6s and B " r5, 7s, the positive interval lower slack sl P K C such that A `sl " B does not necessarily exist.
But we find su " r1, 3s such that A " B ´su.This is, given A and B two non negative closed intervals, with A B, there always exists su or sl in K C such that A " B ´su or A `sl " B, as we proof following.
Proof.On one hand, following Stefanini and Arana [48], if A B, then Ba gH A 0. On the other hand, there always exists Since C " B a gH A 0, then C P K C .In case (a), we have that B " A `C, we define sl " C; consequently, the proposition holds.And in case (b), that is, A " B `p´1qC, we define su " C, and the proof is complete.Based on the previous propositions, we get the following useful result for formulating the forthcoming models.
Corollary 1.Given A, B P K C , then A B if and only if then there exist sl, su P K C such that A `sl " B ´su, with sl " 0 or su " 0.
Proof.It was shown in the proof of Proposition 1 that, if A B, there exists sl P K C or su P K C , depending on the case (a) or (b), and we then have su " 0 or sl " 0 respectively.In the reverse direction, if A`sl " B´su, with sl, su P K C , then, from Proposition 2, it follows that A B.
Remark 1.Given the definition of the gH-difference, Eq.( 3), Corollary 1 actually implies that sl " B a gH A and su " 0, when b ´a ď b ´a (i.e., A `sl " ra, as rb ´a , b ´as " rb, bs " B ´su ), and sl " 0 and su " B a gH A, otherwise (i.e., A `sl " ra, as " rb, bs ´rb ´a , b ´as " B ´su).
Remark 2. In the previous result, if A and B are crisp, with sl " 0 or su " 0, it implies that both sl and su are crisp.
In the next section, we present our proposed model based on applying these previous results to the (IDEA) model ( 4), aiming to unify the two phases (IDEA and PIDEA) and exhausting the constraints to equalities at once.

Enhanced interval slack-based model
Let us recover the (IDEA) model ( 4), and take into account the previous discussions on its constraints, as well as Proposition 1.
Comparing (IDEA) and (EINL), we tighten the input and output constraints to equality, by considering low and upper slack variables on both sides of the constraints such that only one of them can be non-zero (as imposed in Eq. ( 12)).
Moreover, as stated in the following result, the inefficiency measure of any DMU under the (EINL) model is always higher than or equal to the inefficiency measure under (IDEA).
Proposition 3. Given any DMU p , it is verified that EIpX p , Y p q ě IpX p , Y p q.
Proof.For a given DMU p , let ps x ˚, s y ˚, λ ˚q be the optimal solution of (IDEA) (4), and pL x ˚, R x ˚, L y ˚, R y ˚, λ ˚˚q the optimal solution of (5).If we define su x " "


, and su y " 0, from ( 6) and ( 7), it is straightforward that psl x , su x , sl y , su y , λ ˚˚q is a feasible solution for (EINL), and from (4) and ( 8) that EIpX p , Y p q ě IpX p , Y p q.
Given the (EINL) model ( 8), it is expected that efficient DMUs have a null inefficiency measure.We prove that only efficient DMUs satisfy the latter and, hence, that EIpX p , Y p q " 0 is a characterization of efficient DMUs.
Theorem 3. DMU p is efficient if and only if EIpX p , Y p q " 0.
Proof.By contradiction.Let us assume that EIpX p , Y p q " 0.Then, if DMU p is not efficient there exist px ˚, y ˚q P T IDEA such that, x ˚ X p and Y p y ˚.As px ˚, y ˚q P T IDEA , then it also holds that x ˚ ř N j"1 λ j x i j and y ˚ ř N j"1 λ j y r j for some λ ˚P R N `, with ř N j"1 λ j " 1.Then, we have that and Y p ř N j"1 λ j y r j .Applying Corollary 1, we find some sl x ˚, su x ˚ 0, with sl x ˚¨su x ˚" 0, such that ř N j"1 λ j x i j `sl x i " x ip ´su x i , and sl x ˚ 0, or su x ˚ 0. This is due to the constraint derived from the definition of efficiency.Similarly, in the case of the outputs, there are some sl y ˚, su y ˚ 0, with sl y ˚¨su y ˚" 0, such that ř N j"1 λ j y r j ´su  9) and (10), then it belongs to the PPS, px ˚, y ˚q P T, and x ˚ X p and y ˚ Y p , with px ˚, y ˚q pX p , Y p q.This implies a contradiction with the fact that DMU p is efficient.
The previous mathematical program pEINLq is nonlinear, but we can provide an equivalent linear program with 0 ´1 variables leading to the following Enhanced Inefficiency Mix-Integer Linear program (EIMIL), s.t. ( 9) ´( 11), ( 13) ´( 14) where equations ( 16) to (20) are equivalent to the non-linear constrains (12), and y r are real positive constants, and large enough.Recall that a number can be identified with an interval whose extremes are equal and coincide with such a number.Then, it can be compared with intervals via interval inequalities.We can consider L x i " R x i " x ip , and L y r " R y r " maxty rp : r " 1, . . ., Su.Another possibility is to set L x i " R x i " L y r " R y r " maxtx ip , y rp : i " 1, . . ., M, r " 1, . . ., Su.Thus, (EIMIL) is an equivalent mix-integer linear program formulation to (EINL), which computes the same Enhanced Inefficiency measure EIpX p , Y p q.
Let psl x ˚, su x ˚, sl y ˚, sl y ˚, λ ˚, z x ˚, z y ˚q be an optimal solution for pEIMILq model ( 15) for a given DMU p , then we can compute its input and output targets X target p " λ ˚¨X " ř N j"1 λ j X j and Y target p " λ ˚¨Y " ř N j"1 λ j Y j as q is efficient.
Proof.Let psl x ˚, su x ˚, sl y ˚, sl y ˚, λ ˚, z x ˚, z y ˚q be an optimal solution for pEIMILq model ( 15) for a given DMU p .From Eq. ( 9)-( 10) and ( 21)-( 22 q were not efficient there would be some px 1 , y or Y target p y 1 , and ř N j"1 λ j " 1.In summary, we have that In the case of the inputs, applying Corollary 1, we can find some slacks 0 sl x 1 , su x 1 P K C , such that sl x 1 ¨su x 1 " 0, and ř N j"1 λ 1 j X j ´su y 1 " X p `sl y 1 .Analogously, for the output case, we can find some slacks 0 sl y 1 , sl y 1 P K C , such that sl y 1 ¨su y 1 " 0, and ř N j"1 λ 1 j Y j ´su y 1 " Y p `sl y 1 .Moreover, given Remark 1 and that all intervals are K C , it follows / -, and Analogously, for the output case, it holds sl y r " ř N j"1 λ j y r j a gH y rp , su ř N j"1 λ 1 j y r j a gH y rp , , / .

Application to Tourism
This section aims to assess, using data envelopment analysis (DEA), the sustainability efficiency of tourism in the most important Mediterranean regions during 2019.According to the World Tourism Organization, sustainable development is "tourism that takes full account of its current and future economic, social, and environmental impacts, addressing the needs of visitors, the industry, the environment, and host communities."Sustainability is usually represented in three fundamental pillars or dimensions: economic, social, and environmental [38].Sustainability is a recent concept that is very important nowadays for the following reasons: • A key to preserving the planet • It helps to reduce pollution and conserve resources • Creating jobs and stimulating the economy • Improves public health

• Protects biodiversity
• A development that is achievable with political will and public support On the other hand, tourism is considered one of the leading international commerce sectors and one of the primary sources of income for many developing countries.During the last decades, tourism has represented a vital world business and has experienced continued growth.For example, according to World Tourism Organization, international tourism arrivals grew 4,3% in 2014, reaching 1.133 million tourists, and in January-March 2019 compared to the same period last year, below the 6% average growth of the past two years.
However, it is essential to note that 2020 was a challenging year for most sectors cause of the Covid-19 pandemic, and the tourism industry was affected notably.The number of tourist trips undertaken each year before the advent of Covid-19 exceeded the world's population [44].Although according to the latest World Tourism Barometer from UNWTO, this is in the end because international tourism is on track to reach 65% of pre-pandemic levels by the end of 2022, and the sector continues to recover from the pandemic.
Recent studies on sustainable tourism using Data Envelopment Analysis focus on the environmental effects and competitiveness of tourism.For example, Huang et al. [24] use SBM-DEA and Tobit's regression to measure the efficiency of environmental training for diving tourists considering inputs such as education, Diver's qualifications, or length of diving time and output as improper environmental behaviors.Also, regarding eco-efficiency, Li et al.
[35] use two DEA models (CCR and Panel Tobit) to assess the Chinese forest parks in 30 provinces of China, considering inputs as forest park employees, ecological tourism footprint, water consumption or annual forest park tourism data and outputs as total tourism revenue, SO2 emissions or solid particulate emissions.In the matter of evaluating the impact of high-speed rail on the development efficiency of low-carbon in China, Li et al. [36] considering an input (namely, high-speed rail) and an output (namely, low-carbon tourism) through the stochastic production frontier method (SFA) in combination with BCC-DEA models.Bire [8] evaluate Indonesia's Nusa Tenggara Timus province using Malmquist-DEA considering three inputs (namely, number of accommodations, number of restaurants, and number of attractions) and an output (tourist visits) rethinking a new scenario for the regional tourism stakeholders.
Pérez Le ón et al. [43] propose an index for measuring tourist destinations in the Caribbean Region, considering 27 indicators in 4 sub-indexes using DEA and goal programming to build composite indicators and measure the competitiveness of destinations.Flegl et al. [17] measure the hospitality in Mexico using the CCR-DEA model and an input (number of rooms per hotel's star) and three outputs (occupancy rate, tourists arrivals, and related revenue per available room) getting high-efficiency results for national tourism and low-efficiency for international tourism and highlighting that the first is located in land-states and the second in coastal states.

Variables and Data
Several input and output variables are considered from the three sustainable dimensions.See Table 1 and Figure 1.The data refer to the year 2019, prior to the pandemic.We have used the Eurostat database and Regional databases from different Mediterranean regions.We would have liked to include additional regions and variables, but this was prevented by data availability.As a novelty in tourism studies, the variable Bed-places is considered an interval, estimated as a confidence interval from the available data.Thus, Bed-places data come from two databases with different data for some regions.Also, similar to other studies like [20] and [18], GHG emissions have been considered undesirable.
Using interval variables, the proposed DEA model is a novelty in sustainability tourism efficiency assessment.

Results and discussion
The inefficiency scores of the proposed (EIMIL) DEA model EIpX p , Y p q with its corresponding targets and slacks intervals are shown in Table 3.Because Regarding inefficient regions, Nisia Aigaiou (Kriti) needs to increase receipts by around 28% and female and male employment by up to 494% and 447%, re-   spectively, to reach the frontier of tourism sustainability.Regarding the number of beds, the interval variable has an excess of 28% and 12% with respect to their current value according to the Eurostat and the regional databases, respectively.
Similarly, Provence-Alpes-C ôte d'Azur has margins for improvement of 248% and 195% for female and male employment, respectively, and for increasing the receipts by 10%.The interval input variable shows zero slacks for the Eurostat database and around 5% slack for the regional database.Finally, Cyprus is the only study area with a margin of improvement in greenhouse gas emissions.
Namely, they could decrease by 46%.It also has margins of improvement of 246% and 305% in female and male employment, respectively, and of 10% in overnight stays.As for the BP interval input, the slack is zero for the Eurostat database and around 5% for the regional database.
For comparison purposes, Table 3 shows the overall efficiency intervals rΦ L p , Φ U p s computed by the non-oriented SBM model for interval data proposed by Azizi et al. [6].Their method computes an interval measure of efficiency although it does not compute targets.Their approach uses a preference-degree approach for comparing and ranking the DMUs.The corresponding ranking computed by these authors for this dataset is also included in the Table .Their ranking and the proposed approach are not correlated (Spearman correlation coefficient=-0.056).This is undoubtedly due to their considering a double frontier approach.
Figure 2 shows the observed and target input and output targets for these three inefficient DMUs.The values are scaled by the corresponding observed data to facilitate their comparison.Note that, as the fifth output is undesirable, the corresponding targets stayed the same or were reduced.Note also that, in this application, only the input variable involves interval data.

Conclusions
This paper has proposed a new interval-valued DEA approach and associated slacks-based inefficiency measures.It requires solving a mixed-integer linear program that allows computing the corresponding input and output targets.A super-efficiency version of the model has also been formulated in case fully ranking the DMUs is desired.
An application for sustainable tourism efficiency assessment has been presented.The input and output variables span the three sustainability dimensions, including the environmental dimension, represented by GHG emissions from tourism activities.The need to apply an interval DEA approach comes from the fact that for the input variable (Bed-places), the data comes from two different data sources, and in some cases, the corresponding values do not coincide.This is something that often occurs in practice.In order to avoid loss of information, it was decided to represent that variable as an interval using the values from the two sources as limits.The proposed approach has handled this type of variable computing inefficiency scores and targets for all the DMUs.In this application, given the small dataset available, many DMUs were labeled as efficient, which also required solving the corresponding super-efficiency DEA model.
As a continuation of this research, we will apply this approach to other sectors (e.g., healthcare, sports, etc.) where input and output interval data can occur.Theoretically, we could extend it to Network DEA scenarios, i.e., production systems involving multiple interconnected processes.

Acknowledgements
The first and second authors are partially supported by grant PID2019-105824GB-I00.The fifth author acknowledges the financial support of the Spanish Ministry of Science and Innovation, grant PID2021-124981NB-I00.
An approach to measuring sustainable tourism at the local level in Europe.
(i) ra, as rb, bs if and only if a ď b and a ď b. (ii) ra, as ă rb, bs if and only if a ă b and a ă b. (iii) ra, as rb, bs if and only if a ă b and a ď b, or a ď b and a ă b.

Proposition 2 .
If A, B P K C , and sl, su P K C such that A `sl " B ´su, then A B. Proof.Since A A `sl, and B ´su B, then it follows that A A `sl " B ´su B.

Figure 1 :
Figure 1: Input and Outputs considered in this sustainable tourism application.
of the relatively small dataset, only three DMUs are inefficient: Nisia Aigaiou-Kriti, Cyprus, and the Provence-Alpes-C ôte d'Azur.That is why, in order to rank the DMUs fully, the super SBI interval DEA model (S-PEIMIL) has been solved.The corresponding super-inefficiency scores SEIpX p , Y p q and the final ranking are also shown.Note that, regarding sustainable tourism, the two Greek regions considered in this studio have disparate results, with Attiki and Nisia Aigaiou (Kriti) at the top and bottom of the ranking, respectively.The second-best score is for Catalu ña, with similar scores as the Croatian region Jadranska Hrvatska, followed by the Italian regions of Campania and Veneto.On the bottom side, Cyprus and the Provence-Alpes-C ôte d'Azur, in addition to Kriti, are inefficient.

[ 3 ]
Apt, K.R. and Zoeteweij, P.A. Comparative Study of Arithmetic Constraints on Integer Intervals.International workshop on constraint solving and constraint logic programming, 3010 (2004) 1-24 [4] Arana-Jiménez, M. Sánchez-Gil, M.C.Younesi, A. and Lozano, S. Integer interval DEA; an axiomatic derivation of the technology and an additive, slacks-based model.Fuzzy sets and systems, 422 (2021) 83-105.[5] Azadi, M., Jafarian, M., Farzipoor Saen, R. and Mirhedayatian, S.M.A new fuzzy DEA model for evaluation of efficiency and effectiveness of suppliers in sustainable supply chain management context.Computers and Operation Research, 54 (2015) 274-285 [6] Azizi, H., Kordrostami, S. and Amirteimoori, A. Slack-based measure of efficiency in imprecise data envelopment analysis: An approach based on the first case, the DEA technology considers Constant Returns to Scale (CRS), while in the second case, the DEA technology considers Variable Returns to Scale (VRS).Let us also recall that a DMU p is said to be efficient if and only if for any px, yq P T DEA such that x X p and y Y p , then px, yq " pX p , Y p q.
N ÿ j"1 (8)nd sl y ˚ 0, or su y ˚ 0. By construction, it is straightforward that pλ ˚, sl x ˚q, su x ˚, sl y ˚, su y ˚is a feasible solution of model pEINLq, but its objective function(8)is strictly greater than zero, which is a contradiction with EIpX p , Y p q " 0. Now let us suppose DMU p is efficient, but EIpX p , Y p q ą 0. Then there are We can then compute px ˚, y ˚q as x i " x ip for all i P t1, . . ., Mu, i r o P t1, . . ., Su.

Table 1 :
Description and data of the Input and Outputs for the Tourism application

Table 2 :
(15)lts from the (EIMIL)(15)(second column) and (S-EIMIL) (37) (third column) models, respectively.The corresponding ranking is given in the fourth column.We also include the slacks and the targets.Only the inefficient DMUs have non-zero slacks.For clarity, we represent those null interval slacks

Table 3 :
Comparison with other models from the Literature.