Data envelopment analysis for weight derivation and aggregation in ...

Computers & Operations Research 33 (2006) 1289–1307 

www.elsevier.com/locate/cor 

Data envelopment analysis for weight derivation and aggregation 

in the analytic hierarchy process 

Ramakrishnan Ramanathan ∗ 

College of Commerce and Economics, Sultan Qaboos University, Post Box 20, Postal Code 123, Oman 

Available online 11 November 2004 

Abstract 

Data envelopment analysis (DEA) is proposed in this paper to generate local weights of alternatives from pair-wise 

comparison judgment matrices used in the analytic hierarchy process (AHP). The underlying assumption behind 

the approach is explained, and some salient features are explored. It is proved that DEA correctly estimates the 

true weights when applied to a consistent matrix formed using a known set of weights. DEA is further proposed to 

aggregate the local weights of alternatives in terms of different criteria to compute final weights. It is proved further 

that the proposed approach, called DEAHP in this paper, does not suffer from rank reversal when an irrelevant 

alternative(s) is added or removed. 

2004 Elsevier Ltd. All rights reserved. 

Keywords: Analytic hierarchy process; Data envelopment analysis; Aggregation; Independence of irrelevant alternatives 

1. Introduction 

Data envelopment analysis (DEA) is one of the most popular tools in production management literature 

for performance measurement, while the analytic hierarchy process (AHP) is a popular tool in the field 

of multiple-criteria decision-making (MCDM). MCDM has been succinctly defined as making decisions 

in the face of multiple conflicting objectives [1]. Many researchers have found similarities between 

DEA and MCDM techniques. One of the earliest attempts to integrate DEA with multi-objective linear 

programming (a MCDM technique) was provided by Golany [2]. Since then, there have been several 

attempts to use the principles of DEA in the MCDM literature. The traditional goals of DEA and MCDM 

∗ Tel.: +968 513 333 X2849; fax: +968 514 043. 

E-mail address: ramanathan@squ.edu.om (R. Ramanathan). 

0305-0548/$ - see front matter 2004 Elsevier Ltd. All rights reserved. 

doi:10.1016/j.cor.2004.09.020

1290 R. Ramanathan / Computers & Operations Research 33 (2006) 1289–1307 

have been compared by Stewart [3]. The goal of DEA is to determine the productive efficiency of a system 

or decision-making-unit (DMU) by comparing how well the DMU converts inputs into outputs, while the 

goal of MCDM is to rank and select from a set of alternatives that have conflicting criteria. It has been 

recognized for more than a decade that the MCDM and DEA formulations coincide if inputs and outputs 

can be viewed as criteria for performance evaluation, with minimization of inputs and/or maximization 

of outputs as associated objectives. [3–5] 

It is not the intention in this paper to go into the details of similarities. Interested readers are referred to 

recent reviews of multicriteria methods [6,7], books [8,9], and other related articles including those cited 

above [10–12]. This paper is concerned about using the concepts of DEA for addressing specific problems 

in the AHP [13], a popular technique of MCDM. The paper begins with brief discussions on DEA and 

AHP and a literature survey on previous approaches linking the two methods. The proposed approach 

synthesizing DEA concepts in AHP is discussed and illustrated in Section 3. Some of the salient features 

of the proposed approach are discussed in more detail in Section 4. The paper ends with a summary and 

conclusions of the paper. 

2. The methodologies: DEA and AHP 

2.1. Data envelopment analysis 

DEA has been successfully employed for assessing the relative performance of a set of firms, usually 

called the DMU, which use a variety of identical inputs to produce a variety of identical outputs. The 

concept of Frontier Analysis suggested by Farrel [14] forms the basis of DEA, but the recent series of 

discussions started with the article by Charnes et al. [15]. 

Assume that the there are N DMUs producing J outputs using I inputs. Let the mth DMU produce 

outputs ymj using xmi inputs. The resulting output–input structure of DMUs is shown in Table 1. The 

objective of the DEA exercise is to identify the DMU that produces the largest amounts of outputs by 

consuming the least amounts of inputs. This DMU (or DMUs) is considered to have an efficiency score 

equal to one. The efficiencies of other inefficient DMUs are obtained relative to the efficient DMUs, and are 

assigned efficiency scores between zero and one. The efficiency scores are computed using mathematical 

programming. 

Since DEA is now a widely recognized technique, it is not described in this paper. Interested readers 

are referred to more detailed materials [8,16–18]. Four basic DEA models (Models 1–4) are presented 

below. These models can be used to calculate the DEA efficiency score of mth DMU. Some of these 

models will be used in the analyses later in this paper. The optimal objective function values of Models 

1 and 3, when solved, represent the efficiency score of the mth DMU. This DMU is relatively efficient if 

and only if their optimal objective function value equals unity. As informed earlier, efficiency scores for 

inefficient units are between zero and one. For inefficient units, DEA also provides those efficient units 

(namely peers), which the inefficient units can emulate to register performances that could improve their 

efficiency scores. 

Because of the nature of formulations, the optimal objective function values of Models 2 and 4 represent 

the reciprocal of efficiency scores. Models 1–4 assume constant returns to scale (CRS) which is said to 

prevail when an increase of all inputs by 1% leads to an increase of all outputs by 1% [19]. Models 1 and 

3 are said to be input-oriented as the objective is to produce the observed outputs with a minimum input

Table 1 

DMUs, outputs and inputs for a DEA model 

R. Ramanathan / Computers & Operations Research 33 (2006) 1289–1307 1291 

Output 1 Output 2 Output J Input 1 Input 2 Input I 

DMU 1 y11 y12 … y1J x11 x12 … x1I 

DMU 2 

. 

y21 

. 

y22 

. 

… 

. 

y2J 

. 

x21 

. 

x22 

. 

… 

. 

x2I 

. 

DMU N YN1 yN2 … yNJ xN1 xN2 … xNI 

level [20]. Models 2 and 4 are said to be output-oriented. More complex DEA formulations, including the 

variable returns to scale (VRS) and Archimedean infinitesimals, are also available in literature [8,16,17]. 

In all these models, decision variables are the values of the multipliers (elements of matrices U and V 

for Model 1, U ′ and V ′ for Model 2, for Model 3, and for Model 4). Scalars and are also decision 

variables for Models 3 and 4. The subscript m denotes the DMU for which efficiency computations are 

made. X and Y are the matrices of inputs and outputs, respectively, for all the DMUs, while Xm and Ym 

are the matrices of inputs and outputs, respectively, for the mth DMU. 

Model 1 

Model 2 

Model 3 

Model 4 

max 

U,V 

z = V T m Ym 

such that U T m Xm = 1, 

min 

U ′ ,V ′ 

V T mY − U T mX 0, 

V T m ,UT m 0, 

z ′ = U ′ m T Xm 

such that V ′ m T Ym = 1, 

V ′ m T Y − U ′ m T X 0, 

V ′ m T ,U ′ m T 0, 

min m 

, 

such that Y Ym, 

XmXm, 

0; m free, 

max 

, 

m such that Y mYm, XXm, 

0; m free. 

(1) 

(2) 

(3) 

(4)


2.2. The analytic hierarchy process 

Goal 

C 1 C 2 C 3 C 4 

A 1 A 2 A 3 

Fig. 1. A simple AHP model. 

AHP is one of the most popular MCDM tools for formulating and analyzing decisions. The technique 

is employed for ranking a set of alternatives or for the selection of the best in a set of alternatives. The 

ranking/selection is done with respect to an overall goal, which is broken down into a set of criteria. A 

brief discussion of AHP is provided in this section. More detailed description of AHP and application 

issues can be found in Saaty [13]. AHP has been applied to numerous practical problems in the last few 

decades [21]. 

Application of AHP to a decision problem involves four steps [22]. 

Step 1: Structuring of the decision problem into a hierarchical model. It includes decomposition of 

the decision problem into elements according to their common characteristics and the formation of a 

hierarchical model having different levels. A simple AHP model (Fig. 1) has three levels (goal, criteria 

and alternatives). Though the simple model with three levels shown in Fig. 1 is the most common AHP 

model, more complex models containing more than three levels are also used in the literature. For example, 

criteria can be divided further into sub-criteria and sub-sub-criteria. Additional levels containing different 

actors relevant to the problem under consideration may also be included in AHP studies [13]. 

Step 2: Making pair-wise comparisons and obtaining the judgment matrix. In this step, the elements 

of a particular level are compared with respect to a specific element in the immediate upper level. The 

resulting weights of the elements may be called the local weights (to be contrasted with final weights, 

discussed in Step 4). 

The opinion of a decision-maker (DM) is elicited for comparing the elements. Elements are compared 

pair-wise and judgments on comparative attractiveness of elements are captured using a rating scale (1–9 

scale in traditional AHP [13]). Usually, an element receiving higher rating is viewed as superior (or more 

attractive) compared to another one that receives a lower rating. The comparisons are used to form a 

matrix of pair-wise comparisons called the judgment matrix A. Each entry aij of the judgment matrix are 

governed by the three rules: aij > 0; aij =1/aji; and aii =1 for all i. If the transitivity property holds, i.e., 

aij = aik ∗ akj , for all the entries of the matrix, then the matrix is said to be consistent. If the property does 

not hold for all the entries, the level of inconsistency can be captured by a measure called consistency 

ratio [13] (see next step). 

For the model shown in Fig. 1, five judgment matrices need to be elicited from DM—one for estimating 

the local weights of criteria with respect to the goal (see Table 2A for an illustrative judgment matrix), and 

one each for computing the local weights of alternatives with respect to each of the four criteria (Table 

2B–E). All the entries in Table 2A–E, except the local weights, are hypothetical values.


Table 2 

Illustration of AHP calculations for the model shown in Fig. 1 

A: Comparison of criteria with respect to goal 

C1 C2 C3 C4 Local weights 

C1 1 1 4 5 0.400 

C2 1 1 5 3 0.394 

C3 1/4 1/5 1 3 0.128 

C4 1/5 1/3 1/3 1 0.078 

Consistency ratio = 0.088 

B: Comparison of alternatives with respect to C1 

A1 A2 A3 Local weights 

A1 1 1/3 5 0.279 

A2 3 1 7 0.649 

A3 1/5 1/7 1 0.072 


C: Comparison of alternatives with respect to C2 


A1 1 1/9 1/5 0.060 

A2 9 1 4 0.709 

A3 5 1/4 1 0.231 


D: Comparison of alternatives with respect to C3 


A1 1 2 5 0.582 

A2 1/2 1 3 0.309 

A3 1/5 1/3 1 0.109 


E: Comparison of alternatives with respect to C4 


A1 1 3 9 0.692 

A2 1/3 1 3 0.231 

A3 1/9 1/3 1 0.077 


F: Final weights of alternatives 

A1 

A2 

A3 

0.261 

0.590 

0.148


Step 3: Local weights and consistency of comparisons. In this step, local weights of the elements are 

calculated from the judgment matrices using the eigenvector method (EVM). The normalized eigenvector 

corresponding to the principal eigenvalue of the judgment matrix provides the weights of the corresponding 

elements. Though EVM is followed widely in traditional AHP computations, other methods are also 

suggested for calculating weights, including the logarithmic least-square technique (LLST) [23,24], goal 

programming [25], and others [26]. When EVM is used, consistency ratio (CR) can be computed. For a 

consistent matrix CR = 0. A value of CR less than 0.1 is considered acceptable because human judgments 

need not be always consistent, and there may be inconsistencies introduced because of the nature of scale 

used. If CR for a matrix is more than 0.1, judgments should be elicited once again from the DM till he 

gives more consistent judgments. Local weights computed for the illustrative judgment matrices using 

EVM and the corresponding CR values are also shown in Table 2A–E. 

Step 4: Aggregation of weights across various levels to obtain the final weights of alternatives. Once 

the local weights of elements of different levels are obtained as outlined in Step 3, they are aggregated 

to obtain final weights of the decision alternatives (elements at the lowest level). For example, the final 

weight of alternative A1 is computed using the following hierarchical (arithmetic) aggregation rule in 

traditional AHP: 

Final weight of A1 = 

 

Local weight of A1 with Local weight of 

× 

. (5) 

respect of criterion Cj criterion Cj 

j 

By definition, the weights of alternatives and importance of criteria are normalized so that they sum to 

unity. 

Note that a geometric variant of the above aggregation rule can also be used, especially when LLST 

is used for computing local weights [24]. The final weights represent the rating of the alternatives in 

achieving the goal of the problem. 

The final weights computed using (5) for the illustration are shown in Table 2F. Thus, given the 

hypothetical judgments of Table 2A–E, alternative A2 is the most preferred alternative, followed by 

alternatives A1 and then A3. 

The popularity of AHP stems from its simplicity, flexibility, intuitive appeal and its ability to mix 

quantitative and qualitative criteria in the same decision framework. Despite its popularity, some shortcomings 

of AHP have been reported in the literature, which have limited its applicability. The number 

of judgements to be elicited in AHP increases as the number of alternatives and criteria increase. This is 

often a tiresome and exerting exercise for the DM. The issue of rank reversal [27] is one of the prominent 

limitations of traditional AHP. The ranking of alternatives determined by the traditional AHP may be 

altered by the addition or deletion of another alternative for consideration. For example, when a new 

alternative A4 is added to the list of alternatives discussed earlier, or when an existing alternative A1, A2 

orA3 is removed, it is possible that their rankings change. Modifications to the traditional AHP have been 

suggested [24] to avoid the problem of rank reversal. 

2.3. A literature survey on linking DEA and AHP 

Both DEA and AHP are versatile tools in their own fields. While DEA has traditionally found applications 

for performance measurement of DMUs, AHP has been widely applied in MCDM problems 

for estimating weights of alternatives when several criteria, both qualitative and quantitative, have to be 

considered. In fact, many modern complex issues have been tackled using a combination of both the


methods. Some articles have used AHP for handling subjective factors and to generate a set of numerical 

values, and then used DEA to identify efficiency score based on the entire data, including those generated 

by the AHP [28,29]. Sinuany-Stern et al. [30] have applied DEA on pairs of units, used the resulting DEA 

scores to generate a pair-wise comparison matrix and then applied AHP to generate weights of units from 

the matrix. 

AHP has also been used to introduce preference information in DEA calculations. Preference information 

is introduced in the form of subjective weights derived using AHP [31,32]. Both these studies have 

incorporated AHP weights in DEA using the method of assurance region [33]. 

Some papers have applied both the methods for a given problem and compared their outputs [34–36]. 

In addition, DEA and AHP have been linked with other techniques for specific applications (e.g., DEA 

with discriminant analysis and goal programming [37], AHP with goal programming [38], and AHP 

with compromise programming [39]). In this paper, the concepts of efficiency measurement in DEA are 

integrated with the concepts of weight measurement in AHP. 

3. A synthesis of concepts of DEA in deriving weights in AHP 

In this section, it is proposed that DEA concepts can be used in the last two steps of applying AHP to a 

decision problem—namely, deriving local weights from a given judgment matrix (Step 3) and aggregating 

local weights to get final weights (Step 4). 

3.1. Using DEA concepts for deriving local weights from a judgment matrix 

In this section, we propose DEA for deriving local weights from a judgment matrix. Efficiency calculations 

using DEA require outputs and inputs. Each row of the judgment matrix is viewed as a DMU and 

each column of the judgement matrix is viewed as an output. Thus a judgement matrix of size n × n will 

have n DMUs and n outputs. Note that the entries of the matrix are viewed as outputs as they have the 

characteristics of outputs, viz. an element having higher rating is viewed better than the one that has a 

lower rating. Since DEA calculations cannot be made entirely with outputs and require at least one input, 

a dummy input that has a value of 1 for all the DMUs is employed. A comparison of traditional AHP 

view and the proposed DEA view of a judgement matrix is shown in Fig. 2. It is proposed in this paper 

that the efficiency scores calculated using DEA models such as Models 1–4 could be interpreted as the 

local weights of the DMUs. 

3.1.1. Local weights using DEA for consistent judgment matrices 

When applied to a consistent matrix, for which weights are known, DEA correctly estimates the true 

weights. Specifically, suppose that a predetermined set of n weights, say w1,w2,...,wn, is used to create 

a consistent judgment matrix of size n given by 

⎡ 

w1/w1 w1/w2 ··· 

⎤ 

w1/wn 

⎢ w2/w1 

W = ⎢ 

⎣ 

. 

w2/w2 

. 

··· 

. 

w2/wn ⎥ 

. 

⎦ 

wn/w1 wn/w2 ··· wn/wn 

.


Element 1 

Traditional AHP View Proposed DEA View 

Element 2 

… 

Element n 

Element 1 1 a 12 … a 1N 

Element 2 1/ a 12 1 … a 2N 

… … … … … 

Element N 1/ a 1 N 1/ a 2 N … 1 

Output 1 

Output 2 

… 

Output n 

DMU 1 1 a 12 … a 1N 

DMU 2 1/ a 12 1 … a 2N 1 

Dummy input 

… … … … … … 

DMU N 1/ a 1N 1/ a 2 N … 1 1 

Fig. 2. A comparison of the traditional AHP view and the proposed DEA view of a judgement matrix. 

Any weight derivation method, when applied to the above matrix W, should generate the true weights 

w1,w2,...,wn. DEA correctly calculates the true weights for a consistent judgment matrix as proved in 

Theorem 1. Note that it is not possible to make a similar verification for inconsistent matrices, because 

there is no unique correct answer. 

Theorem 1. For a consistent judgment matrix, the local weights of alternatives computed using DEA 

coincide with the correct weights used to form the matrix. 

Proof. Let there be n elements for comparison, leading to a judgment matrix W of size n. Without loss 

of generality, assume that w1 w2 ···wn where wi represents the true weight of element i. 

Model 4, with one dummy constant input, for computing the weight of element m is the following: 

Model 6 

max m 

such that 

n wi 

k mi, i=1 wm 

n 

mi 1, 

i=1 

0; m free, 

where are the multipliers. Subscript m represents the element for which the efficiency is being computed. 

Note that, because of the nature of the consistent judgment matrix, the set of constraints Y mYm in 

Model 4 reduces to a single constraint m n i=1 (wi/wm)mi in Model 6. Again because of the presence 

of single constant dummy input, the second set of constraints XXm in Model 4 reduces to a single 

constraint n i=1mi 1 in Model 6. Further, it has been proved that this constraint reduces to strict equality 

ni=1 

mi = 1 at the optimal solution [40]. 

Substitute m = n in Model 6 to compute the weight of the last element n. At the optimal solution, 

∗ nn = 1 − n−1 i=1 ∗ where asterisk indicates optimal solution. Substituting in the first constraint of 

ni 

1 

(6)


Table 3 

A look at the entries of Table 2A from the DEA perspective 

DMU Output 1 Output 2 Output 3 Output 4 Input (dummy) DEA efficiency 

C1 1 1 4 5 1 1 

C2 1 1 5 3 1 1 

C3 1/4 1/5 1 3 1 0.6 

C4 1/5 1/3 1/3 1 1 0.333 

Model 6, we get 

⎧ 

∗ n 

w1 

wn 

⎪⎨ 

∗ w2 

n1 + wn ∗ wn−1 

n2 +···+ wn ∗ n,n−1 

w1 

wn ∗ w2 

n1 + wn ∗ wn−1 

n2 +···+ wn ∗ n,n−1 + 

⎪⎩ 

+ wn 

wn ∗ nn , 

 

w1 1 + − 1 wn ∗ n1 + 

 

w2 − 1 wn ∗ n2 +···+ 

 

1 − n−1 

 

, 

 

i=1 

∗ ni 

 

wn−1 − 1 wn ∗ n,n−1 . 

If w1 w2 ···wn, all the entries in square brackets in the last equation of (7) will be negative. Hence, 

the maximum value of ∗ n occurs when all the ∗ are at their minimum, i.e., when ∗ n1 =∗ n2 =···=∗ n,n−1 =0. 

Hence the optimal solution is ∗ n = 1, ∗ nn = 1 and ∗ = 0, i = 1, 2,...,n− 1. 

ni 

Using a similar logic, it can be proved that ∗ 1 = wn/w1, ∗ 2 = wn/w2,..., ∗ n−1 = wn/wn−1. Due 

to the nature of the output-oriented envelopment DEA formulation, weights are actually the reciprocals 

of . Thus, the weights of the elements are in the ratio [w1/wn,w2/wn,...,wn−1/wn, 1] or 

[w1,w2,...,wn−1]. 

3.1.2. Numerical illustration 

As an example, consider Table 2A, which is a judgment matrix for comparing criteria with respect to 

the goal. In the proposed approach, the entries of Table 2A are viewed as the performance (row-wise) of 

DMUs C1, C2, C3 and C4 in terms of four outputs. The output–input structure of DEA is shown in Table 

3. In this case, any of Models 1–4 can be applied to get the local weights of the criteria. For example, to 

get the final weight of criterion C1, the following model (based on Model 1) can be used: 

Model 8 

max z = v11 + v12 + 4v13 + 5v14 

s.t. u11 = 1, 

v11 + v12 + 4v13 + 5v14 − u11 0, 

v11 + v12 + 5v13 + 3v14 − u11 0, 

1/4v11 + 1/5v12 + v13 + 3v14 − u11 0, 

1/5v11 + 1/3v12 + 1/3v13 + v14 − u11 0, 

v11,v12,v13,v14,u11 0. 

(7) 

(8)


Traditional AHP View Proposed DEA View 

Criterion 1 

Criterion 2 

… 

Alternative 1 y 11 y 12 … y 1J 

Alternative 2 y 21 y 22 … y 2J 

Criterion J 

… … … … … 

Alternative N y N1 y N2 … y NJ 

Output 1 

Output 2 

… 

Output J 

DMU 1 y 11 y 12 … y 1J 1 

DMU 2 y 21 y 22 … y 2J 1 

… … … … … … 

DMU N y N1 y N2 … y NJ 1 

Fig. 3. A comparison of the traditional AHP view and the proposed DEA view of the matrix of local weights of alternatives with 

respect to all criteria. 

In the above model, the first subscript (i.e., 1) refers to the reference DMU for which efficiency is 

being computed (criterion C1 here). The optimal objective function value of Model 8, when solved, will 

give the local weight of criterion C1. To get the final weight of other criteria, models similar to Model 8 

should be solved by changing the objective function. The DEA efficiency scores representing the local 

weights of DMUs (i.e., the criteria here) are presented in the last column of Table 3 (1.000, 1.000, 0.600, 

and 0.333). 

Weights computed using the DEA approach differs from those obtained using EVM (see Table 2A). 

Note that, for comparing with the weights computed by DEA, the local weights obtained using EVM 

should be adjusted by dividing all the weights by the largest one, resulting in the weights 1, 0.985, 0.319, 

and 0.195. These weights are different from those calculated by DEA (last column of Table 3). The 

difference is because of the assumptions underlying the methods. 

Note that while DMU C1 “produces” more Output 4, DMU C2 “produces” more Output 3 and hence it 

is not possible to say which one is better than the other. However, DMUs C1 and C2 have produced more 

or equal amounts of outputs compared to DMU C3 while all the DMUs produced more or equal amounts 

of outputs compared to DMU C4. It is not possible to conclusively say that the DM prefers C1 or C2,but 

it is possible to conclusively say that he prefers C1 and C2 to C3, and, C1, C2 and C3 to C4. This leads 

to a situation to believe that the DM would rank C1 and C2 at the same level, then rank C3 and rank C4 

at the bottom. Thus DEA assigns in unit efficiency scores for C1 and C2, a lower score for C3 (0.6) and 

the lowest score for C4 (0.333). 

3.2. Aggregation of local weights to get final weights 

In this section, we propose DEA to aggregate local weights to get final weights of elements (Step 4 of 

AHP). Suppose the local weights of alternatives in terms of all the criteria are available as shown in Fig. 

3. The entry ymj represents the local weight of alternative m with respect to criterion j. In traditional AHP, 

the arithmetic aggregation rule (5) is employed for aggregation. DEA is proposed in this paper for the 

aggregation. When DEA is used, alternatives are considered as DMUs and their local weights in terms of 

Dummy input


the criteria will be viewed as outputs. With a dummy input, the DEA view of the matrix of local weights 

is also shown in Fig. 3. 

Normally, when local weights are aggregated to final weights, the importance measures of criteria 

(local weights of criteria in this case) are also used. For example, the aggregation rule (5) is weighted 

arithmetic aggregation incorporating the local weights of weights. However, DEA does not normally 

require the local weights of criteria for aggregation. Hence, aggregation using DEA could be discussed 

under two cases: (a) without considering the local weights of criteria for aggregation and (b) considering 

local weights of criteria. 

Case (a): When DEA is used for aggregation, the importance measures of criteria are automatically 

generated by DEA as the values of multipliers using linear programming. In this case, a simple DEA 

model (any of models 1–4) of data in Fig. 3 could be used to get the final weights of alternatives. Hence, 

the local weights of criteria are not necessary in DEA. 

Case (b): If needed, the importance measures of criteria can be imposed in DEA using the method of 

assurance region [33]. This is done by appending additional constraints that specify relationships among 

the multipliers in the original DEA model. For example, the importance of criteria are incorporated in 

Model 1 in the form of multipliers vm1 = dj vmj (for all j = 1, 2,...,J and d1 = 1). If criterion 1 is half 

and thrice as important as criteria 2 and 3, respectively, then d2 = 1/2 and d3 = 3. An interesting theorem 

for this Case (b) is proved in the next section. 

The above aggregation procedures assume a simple AHP model. The procedures could be extended 

for more complex models having more levels, but we consider the simple model in the remainder of this 

section. 

3.2.1. Aggregation to final weights using DEA when relationships among importance of criteria are 

exogenously specified 

When the importance of criteria are exogenously specified and incorporated in a DEA model by 

providing additional constraints restricting the values of multipliers as in Case (b) above, the final weights 

of alternatives estimated by DEA is proportional to the weighted sum of local weights as shown in Theorem 

2. 

Theorem 2. Let the local weights of alternatives with respect to different criteria be given by 

⎡ 

⎢ 

⎣ 

. 

y11 y12 ··· y1J 

y21 y22 ··· y2J 

. ··· 

yN1 yN2 ··· yNJ 

. 

⎤ 

⎥ 

⎦ , 

where ymj is the local weight of alternative m with respect to criterion j. There are N alternatives and 

J criteria. If the importance of criteria are incorporated in the form of multipliers vm1 = dj vmj (for all 

j = 1, 2,...,J and d1 = 1) then final weights aggregated using DEA is proportional to the weighted sum 

Jj=1 dj ymj for alternative m. 

Proof. We use the formulation shown by Model 1 for the proof here. For computing the final weight of 

the alternative m Model 1 can be written as follows:


Model 9 

max 

subject to 

J 

vmj ymj 

j=1 

um1 = 1; 

J 

vmj ynj − um1 0; n = 1, 2,...,N, 

j=1 

vmj ,um1 0; j = 1, 2,...,J. 

When the importance of criteria are incorporated by introducing additional constraints of the form 

vm1 = dj vmj (for all j = 1, 2,...,J and d1 = 1) in Model 9, the model can be rewritten as follows: 

Model 10 

max vm1 

s.t. 

J 

dj ymj 

j=1 

J 

vm1 

j=1 

vn1 0. 

dj ynj 1; n = 1, 2,...,N, 

Jj=1 Model 10 will choose the value of vm1 such that the largest value of the constraints vm1 dj ynj is 

equal to unity for n = 1, 2,...,N. Since these constraints are the same for all the N DEA formulations 

for finding the efficiency of all the N DMUs, the value of vm1 is the same for all the DEA programs. Thus, 

the efficiency score of the mth DMU is proportional to J j=1dj ymj . 

Corollary 1. If all the criteria have equal weights, then the final weights are proportional simple sum 

Jj=1 ymj . 

Proof. If all the criteria have equal weights, the multipliers have equal value. In this case, dj = 1 for all 

j = 1, 2,...,J. Hence the efficiency score of the mth DMU is proportional to J j=1 dj ymj or J j=1 ymj . 

 

3.2.2. Numerical illustration 

The calculations of the proposed approach can be illustrated using the entries of Table 2B–E. The local 

weights of alternatives with respect to each of the criteria can be computed using DEA as explained in 

Section 3.1. The local weights are shown in Table 4 (columns 2–5). Aggregation using DEA is discussed 

below for the two cases, (a) and (b). 

Case (a): Without using the local weights of criteria: In this case, any of the models (1) to (4) are 

applied for the local weights in Table 4. The local weights are considered as outputs of alternatives, and 

(9) 

(10)


Table 4 

Local and final weights of alternatives computed using the proposed approach 

Local weights of alternatives with Final weights 

respect to individual criteria 

C1 C2 C3 C4 Case (a) Case (b) 

no restrictions with additional restrictions 

on criteria vm1 = vm2; 

weights vm1 = (1/0.6)vm3; vm1 = 3vm4 

A1 0.714 0.111 1.000 1.000 1.000 0.712 

A2 1.000 1.000 0.600 0.333 1.000 1.000 

A3 0.143 0.556 0.200 0.111 0.556 0.346 

a dummy input is introduced. For example, to get the final weight of alternative A1, the following model 

(based on Model 1) can be used: 

Model 11 

max z = 0.714v11 + 0.111v12 + v13 + v14 

s.t. u11 = 1, 

0.714v11 + 0.111v12 + v13 + v14 − u11 0, 

v11 + v12 + 0.6v13 + 0.333v14 − u11 0, 

0.143v11 + 0.556v12 + 0.2v13 + 0.111v14 − u11 0, 

v11,v12,v13,v14,u11 0. 

The optimal objective function value of Model 11, when solved, will give the final weight of alternative 

A1. To get the final weight of other alternatives, models similar to Model 11 should be solved by changing 

the objective function. The resulting final weights of all the three alternatives are shown in Table 4 (1.000, 

1.000, and 0.556). These weights are logical because A1 performs well in terms of Criteria C3 and C4, 

while A2 performs well in terms of Criteria C1 and C2, and hence it is not possible to decide which one 

of them is better. 

Case (b): using the local weights of criteria: In many AHP studies, the importance measures of criteria 

are usually considered while calculating the final weights. Using the values of local weights of criteria 

(see Table 3), the following additional constraints can be introduced in the DEA model that calculates the 

final weight of DMU m: vm1 = vm2; vm1 = (1/0.6)vm3; vm1 = 3vm4 . Specifically, when calculating the 

final weight of alternative A1, the following constraints should be added to Model 11: v11 = v12; v11 = 

(1/0.6)v13; v11 = 3v14. The resulting final weights of alternatives, shown in the last column of Table 4, 

are 0.712, 1.000, and 0.346. As proved in Theorem 2, the final weights are proportional to the weighted 

sum of local weights. For example, for alternative A1, the weighted sum can be calculated as [(0.714 ∗ 

1) + (0.111 ∗ 1) + (1 ∗ 0.6) + (1 ∗ 0.333)] =1.7583. The weighted sum for the three alternatives are 

1.7583, 2.471, and 0.856, which are proportional to 0.712, 1.000, and 0.346. 

These weights again seem logical when compared with the weights obtained using DEA with no 

restrictions on criteria weights. When criteria 3 and 4 are given lower importance, alternative A1 which 

performs well in terms of these two criteria, cannot be regarded as efficient as alternative A2. Hence, the 

DEA scores now result in lower weight for alternative A1. 

(11)


It is suggested that Case (a) should be preferred when calculating final weights of alternatives unless 

there is a strong reason to introduce criteria weights (Case (b)). Even if criteria weights are very important, 

the final weights calculated with and without restrictions (i.e., Cases (a) and (b)) should be presented 

together. 

Because of the synthesis of concepts of DEA with AHP, the new approach proposed in this paper could 

be called “data envelopment analytic hierarchy process (DEAHP)” (thanks to the common “A” in both 

the acronyms). 

4. Some important characteristics of DEAHP 

In this section, some important characteristics of DEAHP are explored in more detail. 

4.1. Independence of irrelevant alternatives 

The rule of independence of irrelevant alternatives (IIR) can be stated as follows. If an alternative is 

eliminated from consideration, then the new ordering for the remaining alternatives should be equivalent 

(i.e., same ordering) to the original ordering for the same alternatives [41]. This issue of IIR is important 

in the context of AHP because it has been reported that traditional AHP does not satisfy this rule [27]. 

DEAHP satisfies this rule as discussed below. 

Note that in DEAHP, the weights of alternatives (i.e., the efficiency scores) are calculated separately 

for each alternative using a separate linear programming model. This can be contrasted from the EVM 

where weights of all the alternatives are derived simultaneously. In addition, while traditional AHP uses 

arithmetic normalization, no such normalization is done in the DEAHP. Further, the DEAHP weights are 

calculated relative to the weight of the best rated alternative. 

For the discussion below, efficient alternatives are interpreted as relevant alternatives because they play 

an important role in the rank ordering of all the alternatives. In a general DEA formulation (which is 

applicable for DEAHP also), if the alternative being eliminated is not an efficient one (i.e., if the alternative 

being eliminated is an irrelevant alternative), then the new ranking calculated will again be relative to 

the highest ranked one, and the ordering of alternatives will not change. This is proved in Theorem 3 

below. For the proof, we use the general DEA input–output matrix shown in Table 1. Since DEAHP is a 

particular case of DEA, the results are applicable to DEAHP also. 

Theorem 3. Consider the input–output matrix shown in Table 1.When DEA is applied to find the efficiency 

scores of the DMUs, the relative rankings of DMUs will not change if an inefficient DMU is removed 

from the original set of DMUs or added to the original set of DMUs. 

Proof. We use the DEA formulation given by Model 1 for the proof. The model maximizes the objective 

function by choosing appropriate values of decision variables V T m , U T m . Note that the objective function 

(i.e., the efficiency score) is 1 for efficient DMUs and less than one for inefficient DMUs. 

Consider the constraints V T mY − U T mX 0. These represent N constraints, one each for the N DMUs. 

First we show that these set of constraints will not be binding (i.e., will not be satisfied in equation form) 

for inefficient DMUs. The proof is given by contradiction. If the constraint is binding for a DMU, say the 

kth DMU, then the corresponding constraint for the kth DMU can be written as V T mYk − U T mXk = 0 which

Table 5 

Data used in Ref. [27]. Source: Ref. [24] 


Criterion Alternatives A1 A2 A3 A4 

A1 1 1/9 1 1/9 

C1 A2 9 1 9 1 

Weight 0.333 A3 1 1/9 1 1/9 

A4 9 1 9 1 

A1 1 9 9 9 

C2 A2 1/9 1 1 1 

Weight 0.333 A3 1/9 1 1 1 

A4 1/9 1 1 1 

A1 1 8/9 8 8/9 

C3 A2 9/8 1 9 1 

weight 0.333 A3 1/8 1/9 1 1/9 

A4 9/8 1 9 1 

leads to V T mYk = 1 when U T mXk = 1. Using the same set of multipliers (V T m ,U T m ) in the DEA formulation 

for kth DMU, we get its efficiency score as unity, which means that the kth DMU has to be efficient. Thus 

for inefficient DMUs the constraint V T mY − U T mX 0 is not binding. 

From the theory of sensitivity analysis in Linear Programming [42], it is obvious that the optimal 

solutions of linear program (Model 1 here) will not change if a non-binding constraint is removed 

or added to the program. Thus the efficiency scores and the relative rankings of DMUs will 

not change if an inefficient DMU is removed from the original set of DMUs or added to the original set 

of DMUs. 

Note that, if the alternative being eliminated is the highest ranked alternative (i.e., if the alternative 

being eliminated is a relevant one), the new ranking of remaining set of alternatives could change. Using 

the logic of Theorem 3, it is clear that constraints given by V T mY − U T mX 0 will be satisfied in equation 

form for efficient DMUs and the theory of LP says that optimal solution of Model 1 may change when 

a binding constraint is removed. The new DEA efficiency scores are calculated relative to the weight 

of the next highest ranked alternative. The efficiency scores of those inefficient alternatives for which 

the removed alternative is a peer will improve while the scores of other inefficient units for which the 

removed alternative is not a peer will not change. 

4.1.1. Illustration 

It can be shown via illustration that DEAHP does not suffer from rank reversal when copies of alternatives 

are added. The famous rank-reversal problem pointed out by Belton and Gear [27] (as reported 

in Ref. [24], see also Ref. [43]) in the context of traditional AHP is used as illustration here. The pairwise 

comparison data used by Belton and Gear [27] is shown in Table 5. Three alternatives, A1, A2, 

and A3, and three criteria with equal weights were initially considered. Note that the entries in all the 

matrices are consistent. The final weights of the alternatives using the traditional AHP are 0.45, 0.47


Table 6 

Final weights of alternatives computed using DEAHP for the data in Table 5 

No restrictions on criteria weights Equal criteria weights 

No copy added Copy of A2 is No copy added Copy of A2 is 

included as A4 

included as A4 

A1 1.000000 1.000000 0.94737 0.94737 

A2 1.000000 1.000000 1.00000 1.00000 

A3 0.200000 0.200000 0.15789 0.15789 

A4 — 1.000000 — 1.00000 

and 0.08, which show that A2 is the alternative with the largest weight. When a copy A4 of A2 is 

added to the set, the final weights calculated using traditional AHP become 0.37, 0.29, 0.06, and 0.29, 

which show A1 as alternative with the largest weight, leading to a reversal in the rank ordering of the 

alternatives. 

The calculations using DEAHP for the same set of data are shown in Table 6. The table shows DEAHP 

calculations when the local weights of alternatives under each criterion is aggregated with no restrictions 

on the weights of criteria, as well as when equal criteria weights are forced. In both the cases the rankings 

are not changed. Because of the absence of normalization, even the numerical values of weights are 

preserved in DEAHP. 

4.2. Some application issues 

Normally, when DEA is applied for computing efficiency scores, two issues are important. 

1. The DMUs should be homogenous units [44]. They should perform the same tasks, and should have 

similar objectives. The inputs and outputs characterizing the performance of DMUs should be identical 

except for differences in intensity or magnitude. AHP is also based on a similar homogeneity 

axiom [45,46]. Hence, the homogeneity requirement is satisfied for the approach proposed in this 

paper. 

2. If the number of inputs and outputs are much larger than the number of DMUs, the discriminating 

power of DEA will be affected. Usually, as the number of inputs and outputs increase, there will be 

more number of DMUs that will get an efficiency rating of 1, as they become too specialized to be 

evaluated with respect to other units. Certain thumb rules are specified in DEA literature to avoid this 

problem. For example, the number of DMUs is expected to be larger than the product of number of 

inputs and outputs or the sample size should be at least 2 or 3 times larger than the sum of the number 

of inputs and outputs [44]. However, there are many examples in the literature where DEA has been 

used with small sample sizes. 

This issue is likely to affect the performance of DEAHP. This may not be very serious when deriving 

local weights using a judgment matrix of size n, as the total number of inputs (one dummy input) and 

outputs (n) is not very large compared to the number of DMUs (n). But, this issue is likely to be serious


when computing the final weights because the total of inputs (1) and outputs (number of criteria) is 

normally smaller than the number of alternatives. This will affect the discriminating power of DEA. 

At the extreme, when all the alternatives receive largest local weights in terms of some criteria, DEA 

will assign unit final weights to all the alternatives. However, note that this is a limitation of DEA in 

the traditional production efficiency applications, but need not be a limitation when applied for deriving 

weights. Any method, including the traditional AHP, will suffer from the same limitation (approximately 

equal final weights for all alternatives) for a similar situation. 

4.3. Computational complexity 

For a single judgment matrix having n alternatives, DEA requires that n different linear programming 

problems are solved to arrive at the weights. Hence, when compared to EVM, DEA requires more 

computational effort. However, given the computational capacity of present-day computers, availability 

of specialized DEA software and the tremendous progress in the computing speeds, this problem will be 

of less concern. 

5. Summary and conclusions 

Data envelopment analysis (DEA) has been proposed in this paper for deriving weights from the 

judgment matrices of the analytic hierarchy process (AHP). If the decision maker is not able, categorically, 

to decide whether one alternative is better than another, he will not be in a position to think that one is 

more important than the other, and this is the logic employed by DEA for calculating the weights. It has 

been proved that DEA calculates true weights for consistent judgment matrices. DEA is further used to 

aggregate local weights of alternatives in terms of different criteria in AHP to final weights. Because of 

the synthesis, the proposed approach is named “data envelopment analytic hierarchy process (DEAHP)”. 

It has been proved and illustrated using a well known example that DEAHP does not suffer from rank 

reversal when an irrelevant alternative(s) is added or removed. Finally, some application issues of DEAHP, 

namely homogeneity, discriminating power and computational efforts, are discussed. Thus, DEAHP has 

been proposed in this paper as an alternative to the traditional methods of weight derivation in AHP, and 

it has been proved that DEAHP can be superior as it can avoid rank-reversal problem in AHP. 

As this paper proposes to combine the concepts in two popular decision making techniques, several 

more issues have to be researched further. For example, it has been proved that DEAHP calculates correct 

weights for a consistent judgment matrix, but its behavior with inconsistent matrices should be studied 

further. This paper has outlined the underlying logic behind the approach and has provided a justification 

for the use of DEAHP. However, the performance of DEAHP when used with matrices with different 

levels of inconsistencies could be empirically studied, similar to the empirical research on comparing the 

traditional eigenvector method of AHP and the logarithmic least square technique [23]. As DEA is based 

on linear programming, DEAHP may be compared and contrasted with other linear programming based 

techniques suggested for deriving weights from judgment matrices, such as goal programming [25,26]. 

Finally, DEAHP seems to be related to some research studies reported in the utility theory literature. Using 

DEA (which aggregates inputs and outputs using a weighted additive function) to derive local weights 

from AHP judgment matrices is closely related to the use of additive value functions in the presence of


partial information about the weights [47,48]. The relationships could be explored further. They form 

topics for further research in this direction. 

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