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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.15 No.1 pp.49-62
DOI : https://doi.org/10.7232/iems.2016.15.1.049

# Using DEA and AHP for Hierarchical Structures of Data

Faculty of Management, Laurentian University, Sudbury, Canada
Corresponding Author, ms_pakkar@laurentian.ca
August 31, 2015 March 15, 2016 March 15, 2016

## ABSTRACT

In this paper, we propose an integrated data envelopment analysis (DEA) and analytic hierarchy process (AHP) methodology in which the information about the hierarchical structures of input-output data can be reflected in the performance assessment of decision making units (DMUs). Firstly, this can be implemented by extending a traditional DEA model to a three-level DEA model. Secondly, weight bounds, using AHP, can be incorporated in the three-level DEA model. Finally, the effects of incorporating weight bounds can be analyzed by developing a parametric distance model. Increasing the value of a parameter in a domain of efficiency loss, we explore the various systems of weights. This may lead to various ranking positions for each DMU in comparison to the other DMUs. An illustrative example of road safety performance for a set of 19 European countries highlights the usefulness of the proposed approach.

## 1.INTRODUCTION

Data envelopment analysis (DEA) is an objective data-oriented 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 decision-maker (DM) out of the decision process.

On the other hand, the analytic hierarchy process (AHP) is a multi-attribute 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:

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; Sinuany-Stern 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:

1. The classical CCR (one-level) 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.

2. A two-level 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).

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

4. A parametric-distance model is developed to explore various sets of weights within the defined domain of efficiency losses.

In two-level 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 sub-categories and further be linked to one another constituting a three-level hierarchical structure. To overcome this limitation, we similarly integrate AHP to a three-level DEA model. A three-level DEA model reflects the characteristics of the generalized multi-level DEA model developed by Shen et al. (2011). Theoretically, the approach proposed in this paper may also be considered as an extension to the three-level 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 unified-scale or normalized data. For this purpose, the distance to a reference approach is adopted as follows (OECD, 2008):

$x i j = x ^ i j x ^ i ( max ) , x ^ i ( max ) = max { x ^ i 1 , x ^ i 2 , ... , x ^ i n } for inputs.$
(1)

$y r j = y ^ r j y ^ r ( max ) , y ^ r ( max ) = max { y ^ r 1 , y ^ r 2 , ... , y ^ r n } for outputs.$
(2)

Where $x ˆ i j$ and $y ˆ r j$ are the raw values of input i(i, 2, …, m) and output r(r =1, 2, …, s) for DMU j(j=1, 2, …, n). xij and yrj 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):

$Max E k = ∑ r = 1 s u r y r k ∑ i = 1 m v i x i k$
(3)
s.t.

$∑ r = 1 s u r y r j ∑ i = 1 m v i x i j ≤ 1 ∀ j ,$
(4)
$u r , v i > 0 ∀ r , i ,$
(5)
where Ek is the relative efficiency of DMU under assessment. k is the index for the DMU under assessment where k ranges over 1, 2, …, n. vi and ur are the weights of input i(i=1, 2, …, m) and output r(r = 1, 2, …, s). The first set of constraints (4) assures that if the computed weights are applied to a group of n DMUs, (j = 1, 2, …, n), they do not attain an efficiency value of larger than 1. The second set of constraints (5) indicates the nonnegative conditions for the model variables.

### 2.2.Three-Level 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 xhhij be the value of input i(i = 1, 2, …, m) of sub-category h′(h′ = 1, 2, …, M′) of category h(h = 1, 2, …, M), and yllrj be the value of output r(r = 1, 2, …, s) of sub-category l′(l′ = 1, 2, …, S′) of category (l = 1, 2, …, S) for DMU j(j = 1, 2, …, n) after normalizing the original data. Let vhhi be the internal weight of input i of sub-category h′ of category h and ullr be the internal weight of output r of sub-category l′ of category l, while $∑ i = 1 m v h h ′ i = 1$ and $∑ r = 1 s u l l ′ r = 1$. Then the values of subcategory h′ of category h and sub-category l′ of category l for the DMU j are defined as $x h h ′ j = ∑ i = 1 m v h h ′ i x h h ′ i j$ and $y l l ′ j = ∑ r = 1 s u l l ′ r y l l ′ r j$, respectively. Let qhh′ be the internal weight of sub-category h′ of category h and pll′ be the internal weight of sub-category l′ of category l, while $∑ h ′ = 1 M ′ q h h ′ = 1$ and $∑ l ′ = 1 S ′ p l l ′ = 1$. Then the values of categories h and l are defined as $x h j = ∑ h ′ = 1 M ′ q h h ′ x h h ′ j$ and $y l j = ∑ l ′ = 1 S ′ p l l ′ y l l ′ j$, respectively. Let qh and pl be the weights of categories h and l, respectively. Then, the new multipliers of input i of sub-category h′ of category h and output r of sub-category l′ of category l are defined as: vhhi = qhqhh′vhhi and ullr = plpll′ullr, respectively. Similarly, the new multiplier of subcategory l′ of category l is defined as: pll = plpll′.

Let $E k ∗$ (k = 1, 2, …, n) be the best attainable efficiency value for the DMU under assessment, calculated from the CCR-DEA model. We want the efficiency value Ek(ullr, vhhi), calculated from the set of weights ullr and vhhi to be closest to $E k ∗$. Our definition of closest is that the largest distance is at its minimum. Hence we choose the form of the minimax model: minullr, vhhi$max k { E k ∗ − E k ( u ′ l l ′ r , v ′ h h ′ i ) }$ to minimize a single deviation which is equivalent to the following nonlinear model:

$Min θ$
(6)
s.t.

$E k ∗ − ∑ l = 1 S ∑ l ′ = 1 S ′ ∑ r = 1 s u ′ l l ′ r y l l ′ r k ∑ h = 1 M ∑ h ′ = 1 M ′ ∑ i = 1 m v ′ h h ′ i x h h ′ i k ≤ θ ,$
(7)
$∑ l = 1 S ∑ l ′ = 1 S ′ ∑ r = 1 s u ′ l l ′ r y l l ′ r j ∑ h = 1 M ∑ h ′ = 1 M ′ ∑ i = 1 m v ′ h h ′ i x h h ′ i j ≤ E j ∗ ∀ j ,$
(8)
$∑ l ′ = 1 S ′ ∑ r = 1 s u ′ l l ′ r = p l ∀ l ,$
(9)
$∑ h ′ = 1 M ′ ∑ i = 1 m v ′ h h ′ i = q h ∀ h ,$
(10)
$∑ r = 1 s u ′ l l ′ r = p ′ l l ′ ∀ l , l ′$
(11)
$∑ i = 1 m v ′ h h ′ i = q ′ h h ′ ∀ h , h ′$
(12)
$u ′ l l ′ r , p ′ l l ′ , p l , v ′ h h ′ i , q ′ h h ′ , q h > 0 ∀ l , l ′ , r , h , h ′ , i ,$
(13)
$0 ≤ θ ≤ 1$
(14)

The combination of (6)-(14) forms a three-level DEA model that identifies the minimum efficiency loss, θ=θmin, 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, $E k ∗$. The second set of constraints (8) satisfies that the efficiency values of all DMUs are less than or equal to their upper bound of $E j ∗$. The sets of constraints (9) to (12) imply that the sum of weights under each (sub-) sub-category equals to the weight of that (sub-) sub-category. It should be noted that the original (or internal) weights of outputs and inputs used for calculating the weighted averages are obtained as ullr = ullr/pll and vhhi = vhhr/qhh while pll = pll/pl and qhh = qhh/ql, respec-tively.

### 2.3.Prioritizing Weights Using AHP

The three-level DEA model identifies the minimum efficiency loss θmin 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 alo(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 1-9 scale.

• Step 2: The AHP method obtains the priority weights of criteria by computing the eigenvector of matrix A (Eq. 15), w=(w1, w2, …,wS)T, which is related to the largest eigenvalue, λmax.

$A w = λ max w .$
(15)

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:

$C . R . = λ max − N ( N − 1 ) R . I .$
(16)
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-) sub-criteria under each (sub-) criterion can be computed. To obtain the weight bounds for output weights in the three-level DEA model, this study aggregates the priority weights of three different levels in AHP as follows:

$u ¯ l l ′ r = w l e l l ′ f l l ′ r , ∑ l = 1 S w l = 1 , ∑ l ′ = 1 S ′ e l l ′ = 1 and ∑ r = 1 s f l l ′ r = 1$
(17)
where wl is the priority weight of criterion l (l=1, …, S) in AHP, ell′ is the priority weight of sub-criterion l′(l′ = 1, 2, ′, S′) under criterion l and fllr is sub-sub-criterion r (r=1, …, s) under sub-criterion l′. Similarly, the weight bounds for input weights in the three-level DEA model, using AHP, is obtained as:

$v ¯ h h ′ i = w ′ h e ′ h h ′ f ′ h h ′ i , ∑ h = 1 M w ′ h = 1 , ∑ h ′ = 1 M ′ e ′ h h ′ = 1 and ∑ i = 1 m f ′ h h ′ i = 1$
(18)
where wh is the priority weight of criterion h (h=1, 2, …, M) in AHP, ehh is the priority weight of sub-criterion h′(h′=1, 2, …, M′) under criterion h and fhhi is sub-subcriterion i(i=1, 2, …, m) under sub-criterion h′.

In order to estimate the maximum efficiency loss θmax necessary to achieve the priority weights of inputs and outputs for each DMU the following sets of constraints are added to the three-level DEA model:

$u ′ l l ′ r = α u ¯ l l ′ r ∀ l , l ′ , r , while α > 0.$
(19)

$v ′ h h ′ i = β v ¯ h h ′ i ∀ h , h ′ , i , while β > 0.$
(20)

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 θminθθmax. Let ullr(θ) and vhhi(θ) be the weights of outputs and inputs for a given value of parameter θ, where outputs are under sub-category l′(l′ =1, 2, …, S′) of category l(l =1, 2, …, S) and inputs are under sub-category h′(h′ =1, 2, …, M′) of category h(h =1, 2, …, M). Let $u ′ l l ′ r ∗$ and $v ′ h h ′ i ∗$ be the priority weights of outputs and inputs obtained from the three-level DEA model after adding (19) and (20). Our objective is to minimize the total deviations of uhhi(θ) and vhhi(θ) from their priority weights, $u ′ l l ′ r ∗$ and $v ′ h h ′ i ∗$ with the shortest Euclidian distance measure subject to the constraints (7) to (13):

Min

$Z k ( θ ) = ( ∑ l = 1 S ∑ l ′ = 1 S ′ ∑ r = 1 s ( u ′ l l ′ r − u ′ l l ′ r ∗ ) 2 + ∑ h = 1 M ∑ h ′ = 1 M ′ ∑ i = 1 m ( v ′ h h ′ i − v ′ h h ′ i ∗ ) 2 ) 1 / 2$
(21)
s.t. Constraints (7) to (13).

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:

$Δ k ( θ ) = Z k ∗ ( θ min ) − Z k ∗ ( θ ) Z k ∗ ( θ min ) ,$
(22)
where $Z k ∗ ( θ )$ is the optimal value of the objective function for θminθθmax. We define Δk (θ) as a measure of closeness which represents the relative closeness of each DMU to its priority weight in the range [0, 1]. Increasing the parameter (θ) , we improve the deviations between the two systems of weights obtained from the three-level DEA model before and after adding the two sets of constraints (19) and (20). This may lead to different ranking positions for each DMU in comparison to the other DMUs. It should be noted that in a special case where the parameter θ = θmax = 0, we assume Δk (θ) = 1.

## 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 DEA-based 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 three-level DEA model for the country under assessment, we obtain an optimal set of weights with minimum efficiency loss (θmin). It should be noted that the efficiency value of all countries calculated from the three-level DEA model is identical to that calculated from the CCR-DEA model. Therefore, the minimum efficiency loss for the country under assessment is θmin = 0 (Table 4). This implies that the measure of relative closeness to the AHP weights for the country under assessment is Δk(θmin)=0. On the other hand, solving the three-level 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, θmax, 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 Δk(θmax) = 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 $v ¯ h h ′ i = v ′ h h ′ i β$ and $u ¯ l l ′ r = u ′ l l ′ r α$ respectively. In addition, the priority weights of safety outcomes in the AHP model can be obtained as follows:

$w ′ h = q h β$ while $∑ h ′ = 1 M ′ ∑ i = 1 m v ′ h h ′ i = q h$ and $∑ i = 1 m v ′ h h ′ i = q ′ h h ′$ for criteria level,

ehh = qhh/qh for sub-criteria level,

fhhi = vhhi/qhh for sub-sub-criteria levels.

Similarly, the priority weights of SPIs in the AHP model can be obtained as follows:

$w l = p l α$ while $∑ l ′ S ′ ∑ r = 1 s u ′ l l ′ r = p l$ and $∑ r = 1 s u ′ l l ′ r = p ′ l l ′$ for criteria level,

ell = pll/pl for sub-criteria level,

fllr = ullr/pll for sub-sub-criteria 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≤θθmax. 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=θmax = 0 we assume Δk(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 θmax 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 three-level 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.

## Figure

The relative closeness to the priority weights of SPIs and safety outcomes [Δ (θ)], versus efficiency loss (θ) for each country.

## Table

Normalized data on SPIs (outputs) and safety outcomes (inputs) for 19 European countries

The AHP hierarchical model for road safety outcomes (inputs)

The AHP hierarchical model for SPIs (outputs)

Minimum and maximum efficiency losses for each country

Optimal weights of SPIs and safety outcomes for the Netherlands obtained from the three-level DEA model bounded by AHP

The ranking position of each country based on the minimum distance to priority weights of SPIs and safety outcomes

The measure of relative closeness to the priority weights of hierarchical SPIs and safety outcomes [Δk(θ)] vs. composite loss [θ] for each country*

*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.

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