1.INTRODUCTION
Data envelopment analysis (DEA) is an objective dataoriented approach for assessing the relative efficiency of decision making units (DMUs) with multiple inputs and outputs. In classical DEA models, each DMU is allowed to choose its own favorable system of weights to maximize its relative efficiency. This freedom from choosing weights is equivalent to keeping the preferences of the decisionmaker (DM) out of the decision process.
On the other hand, the analytic hierarchy process (AHP) is a multiattribute decision making method that can reflect a priori information about the relative priority of inputs, outputs or even DMUs in the efficiency assessment. AHP can be combined with DEA models in different ways. The most common way is the imposition of weight restrictions in DEA models. Referring to the literature, AHP can estimate the bounds of the following restrictions in DEA:

Absolute weight restrictions. These restrictions directly impose upper and (or) lower bounds on the weights of inputs (outputs) using AHP (Pakkar, 2014a; Entani et al., 2004).

Relative weight restrictions. These restrictions limit the relationship between the weights of inputs (outputs) using AHP (Lee et al., 2012; Liu et al., 2005; Takamura and Tone, 2003; Tseng et al., 2009; Kong and Fu, 2012).

Virtual weight restrictions. A single virtual input (output) is defined as the weighted sum of all inputs (outputs). We refer to the proportion of each component of such sum as the “virtual weight” of an input (output). These restrictions limit virtual weights using AHP (Premachandra, 2001; Shang and Sueoshi, 1995).

Restrictions on input (output) units. These restrictions impose bounds on changes of inputs (outputs) while the relative importance of such changes is computed using AHP (Lozano and Villa, 2009).
There are a number of other methods that do not necessarily apply additional restrictions to a DEA model. Such as converting the qualitative data in DEA to the quantitative data using AHP (Azadeh et al., 2008; Ertay et al., 2006; Jyoti et al., 2008; Korpela et al., 2007; Lin et al., 2011; Ramanathan, 2007; Yang and Kuo, 2003; Raut, 2011), ranking the efficient/inefficient units in DEA models using AHP in a two stage process (Ho and Oh , 2010; Jablonsky, 2007; SinuanyStern et al., 2000), weighting the efficiency scores obtained from DEA using AHP (Chen, 2002), weighting the inputs and outputs in the DEA structure (Pakkar, 2014b; Cai and Wu, 2001; Feng et al., 2004; Kim, 2000), constructing a convex combination of weights using AHP and DEA (Liu and Chen, 2004) and estimating missing data in DEA using AHP (Saen et al., 2005). Nevertheless, the above mentioned literature is limited to the applications with onelevel DEA models, which may not entirely satisfy the need for increasingly complex assessment problems.
In a recent paper, Pakkar (2015) proposes an integrated DEA and AHP approach to assess the performance of DMUs. The core logic of the proposed approach is to reflect the relative priority of inputs and outputs in performance assessment under hierarchical structures of data. This approach can be organized into the following steps:

The classical CCR (onelevel) DEA model is used to compute the efficiency of each DMU after normalizing the original data. The computed efficiencies in this step will be part of the data used in the next step.

A twolevel DEA model is used to obtain a set of weights of inputs and outputs for each DMU under the hierarchical structures of data (the minimum efficiency loss).

The two levelDEA model is bounded by AHP weights to reflect the priority weights of inputs and outputs in the performance assessment (the maximum efficiency loss).

A parametricdistance model is developed to explore various sets of weights within the defined domain of efficiency losses.
In twolevel DEA models which originally developed by Meng et al. (2008), the inputs and outputs of similar characteristics are grouped into their own categories using a weighted sum approach. Nonetheless, these inputs and outputs might also belong to different subcategories and further be linked to one another constituting a threelevel hierarchical structure. To overcome this limitation, we similarly integrate AHP to a threelevel DEA model. A threelevel DEA model reflects the characteristics of the generalized multilevel DEA model developed by Shen et al. (2011). Theoretically, the approach proposed in this paper may also be considered as an extension to the threelevel DEA model without explicit inputs, using AHP, to constructing composite indicators proposed by Pakkar (2016).
2.Methodology
2.1.A Basic DEA Model
In a classical DEA model, the optimal values of the variables (weights) are highly sensitive to the scales used for each input and output (Cooper et al., 2004). It seems logical and desirable to have scale independent weights that can be interpreted in some meaningful way. This may be achieved by using a unifiedscale or normalized data. For this purpose, the distance to a reference approach is adopted as follows (OECD, 2008):
Where ${\stackrel{\u02c6}{x}}_{ij}$ and ${\stackrel{\u02c6}{y}}_{rj}$ are the raw values of input i(i, 2, …, m) and output r(r =1, 2, …, s) for DMU j(j=1, 2, …, n). x_{ij} and y_{rj} are the corresponding normalized values of input i and output r for DMU j. Then the fractional CCRDEA model can be developed as follows (Charnes et al., 1978):
2.2.ThreeLevel DEA Model
We develop our formulation based on the generalized distance model (Kao and Hung, 2005) in such a way that the hierarchical structures of data, using a weighted average approach, are taken into consideration (Shen et al., 2011). Let x_{hh′ij} be the value of input i(i = 1, 2, …, m) of subcategory h′(h′ = 1, 2, …, M′) of category h(h = 1, 2, …, M), and y_{ll′rj} be the value of output r(r = 1, 2, …, s) of subcategory l′(l′ = 1, 2, …, S′) of category (l = 1, 2, …, S) for DMU j(j = 1, 2, …, n) after normalizing the original data. Let v_{hh′i} be the internal weight of input i of subcategory h′ of category h and u_{ll′r} be the internal weight of output r of subcategory l′ of category l, while $\sum _{i=1}^{m}{v}_{h{h}^{\prime}i}=1$ and $\sum _{r=1}^{s}{u}_{l{l}^{\prime}r}=1$. Then the values of subcategory h′ of category h and subcategory l′ of category l for the DMU j are defined as ${x}_{h{h}^{\prime}j}={\displaystyle \sum _{i=1}^{m}{v}_{h{h}^{\prime}i}{x}_{h{h}^{\prime}ij}}$ and ${y}_{l{l}^{\prime}j}={\displaystyle \sum _{r=1}^{s}{u}_{l{l}^{\prime}r}{y}_{l{l}^{\prime}rj}}$, respectively. Let q_{hh′} be the internal weight of subcategory h′ of category h and p_{ll′} be the internal weight of subcategory l′ of category l, while $\sum _{{h}^{\prime}=1}^{{M}^{\prime}}{q}_{h{h}^{\prime}}=1$ and $\sum _{{l}^{\prime}=1}^{{S}^{\prime}}{p}_{l{l}^{\prime}}=1$. Then the values of categories h and l are defined as ${x}_{hj}={\displaystyle \sum _{{h}^{\prime}=1}^{{M}^{\prime}}{q}_{h{h}^{\prime}}{x}_{h{h}^{\prime}j}}$ and ${y}_{lj}={\displaystyle \sum _{{l}^{\prime}=1}^{{S}^{\prime}}{p}_{l{l}^{\prime}}{y}_{l{l}^{\prime}j}}$, respectively. Let q_{h} and p_{l} be the weights of categories h and l, respectively. Then, the new multipliers of input i of subcategory h′ of category h and output r of subcategory l′ of category l are defined as: v′_{hh′i} = q_{h}q_{hh′}v_{hh′i} and u′_{ll′r} = p_{l}p_{ll′}u_{ll′r}, respectively. Similarly, the new multiplier of subcategory l′ of category l is defined as: p′_{ll′} = p_{l}p_{ll′}.
Let ${E}_{k}^{\ast}$ (k = 1, 2, …, n) be the best attainable efficiency value for the DMU under assessment, calculated from the CCRDEA model. We want the efficiency value E_{k}(u′_{ll′r}, v′_{hh′i}), calculated from the set of weights u′_{ll′r} and v′_{hh′i} to be closest to ${E}_{k}^{\ast}$. Our definition of closest is that the largest distance is at its minimum. Hence we choose the form of the minimax model: min_{u′ll′r, v′hh′i}${\text{max}}_{k}\left\{{E}_{k}^{\ast}{E}_{k}\left({{u}^{\prime}}_{l{l}^{\prime}r},\text{\hspace{0.17em}}{{v}^{\prime}}_{h{h}^{\prime}i}\right)\right\}$ to minimize a single deviation which is equivalent to the following nonlinear model:
s.t.
The combination of (6)(14) forms a threelevel DEA model that identifies the minimum efficiency loss, θ=θ_{min}, needed to arrive at an optimal set of weights. Constraint (7) ensures that each DMU loses no more than θ of its best attainable efficiency value, ${E}_{k}^{\ast}$. The second set of constraints (8) satisfies that the efficiency values of all DMUs are less than or equal to their upper bound of ${E}_{j}^{\ast}$. The sets of constraints (9) to (12) imply that the sum of weights under each (sub) subcategory equals to the weight of that (sub) subcategory. It should be noted that the original (or internal) weights of outputs and inputs used for calculating the weighted averages are obtained as u_{ll′r} = u′_{ll′r}/p′_{ll′} and v_{hh′i} = v′_{hh′r}/q′_{hh′} while p_{ll′} = p′_{ll′}/p_{l} and q_{hh′} = q′_{hh′}/q_{l}, respectively.
2.3.Prioritizing Weights Using AHP
The threelevel DEA model identifies the minimum efficiency loss θ_{min} needed to arrive at a set of weights of inputs and outputs by the internal mechanism of DEA. On the other hand, the priority weights of inputs and outputs, and the corresponding (sub) categories are defined out of the internal mechanism of DEA by AHP.
In order to more clearly demonstrate how AHP is integrated into the three level DEA model, this research presents an analytical process in which output weights are bounded by the AHP method. All the description on imposing weight bounds for output weights may be easily extendable to input weights. The AHP procedure for imposing weight bounds may be broken down into the following steps:

Step 1: A decision maker makes a pairwise comparison matrix of different criteria, denoted by A, with the entries of a_{lo}(l = o = 1, 2, …,S ). The comparative importance of criteria is provided by the decision maker using a rating scale. Saaty (1980) recommends using a 19 scale.

Step 2: The AHP method obtains the priority weights of criteria by computing the eigenvector of matrix A (Eq. 15), w=(w_{1}, w_{2}, …,w_{S})^{T}, which is related to the largest eigenvalue, λ_{max}.
To determine whether or not the inconsistency in a comparison matrix is reasonable the random consistency ratio, C.R., can be computed by the following equation:
where R.I. is the average random consistency index and N is the size of a comparison matrix. In a similar way, the priority weights of (sub) subcriteria under each (sub) criterion can be computed. To obtain the weight bounds for output weights in the threelevel DEA model, this study aggregates the priority weights of three different levels in AHP as follows:
In order to estimate the maximum efficiency loss θ_{max} necessary to achieve the priority weights of inputs and outputs for each DMU the following sets of constraints are added to the threelevel DEA model:
The two sets of constraints (19)and (20) change the AHP computed weights to weights for the new system by means of two scaling factors α and β . The scaling factors α and β are added to avoid the possibility of contradicting constraints leading to infeasibility or underestimating the relative efficiencies of DMUs (Podinovski, 2004).
2.4.A Parametric Distance Model
We can now develop a parametric distance model for various discrete values of parameter θ such that θ_{min}≤θ≤θ_{max}. Let u′_{ll′r}(θ) and v′_{hh′i}(θ) be the weights of outputs and inputs for a given value of parameter θ, where outputs are under subcategory l′(l′ =1, 2, …, S′) of category l(l =1, 2, …, S) and inputs are under subcategory h′(h′ =1, 2, …, M′) of category h(h =1, 2, …, M). Let ${{u}^{\prime}}_{l{l}^{\prime}r}^{\ast}$ and ${{v}^{\prime}}_{h{h}^{\prime}i}^{\ast}$ be the priority weights of outputs and inputs obtained from the threelevel DEA model after adding (19) and (20). Our objective is to minimize the total deviations of u′_{hh′i}(θ) and v′_{hh′i}(θ) from their priority weights, ${{u}^{\prime}}_{l{l}^{\prime}r}^{\ast}$ and ${{v}^{\prime}}_{h{h}^{\prime}i}^{\ast}$ with the shortest Euclidian distance measure subject to the constraints (7) to (13):
Min
Because the range of deviations computed by the objective function is different for each DMU, it is necessary to normalize it by using relative deviations rather than absolute ones. Hence, the normalized deviations can be computed by:
3.A NUMERICAL EXAMPLE: ROAD SAFETY PERFORMANCE
In this section we present the application of the proposed approach to assess the road safety performance of a set of European countries (or DMUs).The data on 13 road safety performance indicators (SPIs) in terms of road user behavior (inputs) and 4 safety outcomes (outputs) for 19 European countries have been adopted from Shen et al. (2011). The resulting normalized data based on (1) and (2) are presented in Table 1.The notations in Table 1 are as follows: AT = Austria, BE = Belgium, CZ = Czech Republic, DK = Denmark, FR = France, DE = Germany, EL = Greece, HU = Hungary, IE = Ireland, LV = Latvia, LU = Luxembourg, NL = Netherlands, PL = Poland, PT = Portugal, SI = Slovenia, ES = Spain, SE = Sweden, CH = Switzerland, UK = United Kingdom.
Since in DEAbased road safety models, the most efficient countries are those with minimum output levels given input levels, we treat all the outputs (inputs) as inputs (outputs) (Shen et al., 2012). Tables 2 and 3 depict the priority weights of safety outcomes and SPIs as constructed by the author in Expert Choice software. One can argue that the priority weights of SPIs must be judged by road safety experts. However, since the aim of this section is just to show the application of the proposed approach on numerical data, we see no problem to use our judgment alone.
Solving the threelevel DEA model for the country under assessment, we obtain an optimal set of weights with minimum efficiency loss (θ_{min}). It should be noted that the efficiency value of all countries calculated from the threelevel DEA model is identical to that calculated from the CCRDEA model. Therefore, the minimum efficiency loss for the country under assessment is θ_{min} = 0 (Table 4). This implies that the measure of relative closeness to the AHP weights for the country under assessment is Δ_{k}(θ_{min})=0. On the other hand, solving the threelevel DEA model for the country under assessment after adding the two sets of constraints (19) and (20), we adjust the priority weights of SPIs (outputs) and safety outcomes (inputs) obtained from AHP in such a way that they become compatible with the weights’ structure in the three level DEA model. This results in the maximum efficiency loss, θ_{max}, for the country under assessment (Table 4). In addition, this implies that the measure of relative closeness to the AHP weights for the country under assessment is Δ_{k}(θ_{max}) = 1.
Table 5 presents the optimal weights of SPIs and safety outcomes as well as the scaling factors for the best performing country, the Netherlands. It should be noted that the priority weights of AHP used for incorporating weight bounds on the weights of safety outcomes and SPIs are obtained as ${\overline{v}}_{h{h}^{\prime}i}=\frac{{{v}^{\prime}}_{h{h}^{\prime}i}}{\beta}$ and ${\overline{u}}_{l{l}^{\prime}r}=\frac{{{u}^{\prime}}_{l{l}^{\prime}r}}{\alpha}$ respectively. In addition, the priority weights of safety outcomes in the AHP model can be obtained as follows:
${{w}^{\prime}}_{h}=\frac{{q}_{h}}{\beta}$ while $\sum _{{h}^{\prime}=1}^{{M}^{\prime}}{\displaystyle \sum _{i=1}^{m}{{v}^{\prime}}_{h{h}^{\prime}i}={q}_{h}}$ and $\sum _{i=1}^{m}{{v}^{\prime}}_{h{h}^{\prime}i}={{q}^{\prime}}_{h{h}^{\prime}}$ for criteria level,
e′_{hh′} = q′_{hh′}/q_{h} for subcriteria level,
f′_{hh′i} = v′_{hh′i}/q′_{hh′} for subsubcriteria levels.
Similarly, the priority weights of SPIs in the AHP model can be obtained as follows:
${w}_{l}=\frac{{p}_{l}}{\alpha}$ while $\sum _{{l}^{\prime}}^{{S}^{\prime}}{\displaystyle \sum _{r=1}^{s}{{u}^{\prime}}_{l{l}^{\prime}r}={p}_{l}}$ and $\sum _{r=1}^{s}{{u}^{\prime}}_{l{l}^{\prime}r}={{p}^{\prime}}_{l{l}^{\prime}}$ for criteria level,
e_{ll′} = p′_{ll′}/p_{l} for subcriteria level,
f_{ll′r} = u′_{ll′r}/p′_{ll′} for subsubcriteria levels.
Going one step further to the solution process of the parametric distance model (21), we proceed to the estimation of total deviations from the AHP weights for each country while the parameter θ is 0≤θ≤θ_{max}. Table 6 represents the ranking position of each country based on the minimum deviation from the priority weights of SPIs and safety outcomes for θ = 0. It should be noted that in a special case where the parameter θ=θ_{max} = 0 we assume Δ_{k}(θ)=1.
Table 6 shows that the Netherlands is the best performer in terms of the efficiency value and its relative closeness to the priority weights of SPIs and safety outcomes. Nevertheless, increasing the value of θ from 0 to θ_{max} has two main effects on the performance of the other countries: improving the degree of deviations and reducing the value of efficiency. This, of course, is a phenomenon, one expects to observe frequently. The graph of Δ(θ) versus θ, as shown in Figure 1, is used to describe the relation between the relative closeness to the priority weights of SPIs and safety outcomes, versus efficiency loss for each country. This may result in different ranking positions for each country in comparison to the other countries (Appendix A).
4CONCLUSION
We develop an integrated approach based on DEA and AHP methodologies for hierarchical structures of inputs and outputs. We define two sets of weights of inputs and outputs in a threelevel DEA framework. The first set represents the weights of inputs and outputs with minimum efficiency loss. The second set represents the corresponding priority weights of hierarchical inputs and outputs, using AHP, with maximum efficiency loss. We assess the performance of each DMU in comparison to the other DMUs based on the relative closeness of the first set of weights to the second set of weights. Improving the measure of relative closeness in a defined range of efficiency loss, we explore the various ranking positions for the DMU under assessment in comparison to the other DMUs. To demonstrate the effectiveness of the proposed approach, an illustrative example of road safety performance of a set of 19 European countries is carried out.