Computers & Operations Research 32 (2005) 2235–2254www.elsevier.com/locate/dsw

Linear programming models for estimating weights in theanalytic hierarchy process

Bala Chandrana, Bruce Goldenb;∗, Edward Wasilc

aDepartment of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, USAbR.H. Smith School of Business, University of Maryland, College Park, MD 20742, USA

cKogod School of Business, American University, Washington, DC 20016, USA

Abstract

We present an approach based on linear programming (LP) that estimates the weights for a pairwisecomparison matrix generated within the framework of the analytic hierarchy process. Our approach makessense for a number of reasons, which we discuss. We apply our LP approach to several sample problemsand compare our results to those produced by other, widely used methods. In addition, we extend our linearprogram to include applications where the pairwise comparison matrix is constructed from interval judgments.? 2004 Elsevier Ltd. All rights reserved.

Keywords: Analytic hierarchy process; Linear programming; Interval AHP; Sensitivity analysis

1. Introduction

In the late 1970s, Saaty [1,2] developed the analytic hierarchy process (AHP) as a robust approachto multicriteria decision making. In the last 25 years, the AHP has been applied in more than 30diverse areas to rank, select, evaluate, and benchmark decision alternatives (see [3,4]).In the AHP, the decision maker models a problem as a hierarchy of criteria, subcriteria, and

alternatives. After the hierarchy is constructed, the decision maker assesses the importance of eachelement at each level of the hierarchy. This is accomplished by generating entries in a pairwisecomparison matrix where elements are compared to each other. For each pairwise comparison matrix,the decision maker typically uses the eigenvector method (EM) (more about this method in the nextsection) to generate a priority vector that gives the estimated, relative weights of the elements ateach level of the hierarchy. Weights across various levels of the hierarchy are then aggregated usingthe principle of hierarchic composition to produce a ;nal weight for each alternative.

∗ Corresponding author. Tel.: +1-301-405-2232; fax: +1-301-405-3364.E-mail address: bgolden@rhsmith.umd.edu (B. Golden).

0305-0548/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2004.02.010

2236 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

In this paper, we present a novel approach based on linear programming (LP) to generate a priorityvector within the framework of the AHP. The rest of this paper is organized as follows. In Section2, we review two common methods for deriving priority vectors. In Sections 3 and 4, we formulateour linear program for deriving priority vectors, discuss the advantages of our LP approach, andpresent possible extensions, in particular, to the recently studied interval AHP. In Section 5, weapply the LP approach to ;ve pairwise comparison matrices and discuss the results. In Section 6,we present our conclusions.

2. Estimating weights: traditional methods

Over the years, several methods have emerged for estimating the weights from a matrix of pairwisecomparisons including EM and logarithmic least squares (LLS). A discussion of the advantages ofthe competing methods is provided by Harker and Vargas [5].EM was developed by Saaty [2] and is the most widely used method. The popular AHP software

package Expert Choice [6] uses EM to generate priority vectors. EM solves an eigenvalue problemassociated with an n× n pairwise comparison matrix in the following way.Let A= (aij) for all i; j = 1; 2; : : : ; n denote a square pairwise comparison matrix, where aij gives

the importance of element i relative to element j. Each entry in matrix A is positive (aij ¿ 0) andreciprocal (aij = 1=aji for all i; j = 1; 2; : : : ; n). The decision maker wants to compute a vector ofweights (w1; w2; : : : ; wn) associated with A.If the matrix A is consistent (that is, aij = aikakj for all i; j; k = 1; 2; : : : ; n), then A contains no

errors (the weights are already known) and we have

aij = wi=wj; i; j = 1; 2; : : : ; n: (1)

Summing over all j, we obtainn∑

j=1

aijwj = nwi; i = 1; 2; : : : ; n (2)

which, in matrix notation, is equivalent to

Aw = nw: (3)

The vector w is the principal right eigenvector of the matrix A corresponding to the eigenvalue n.If the vector of weights is not known, then it can be estimated from the pairwise comparison

matrix A generated by the decision maker by solving

Aw = �w (4)

for w. The matrix A contains the pairwise judgments of the decision maker and approximates thematrix A whose entries are unknown. In (4), � is an eigenvalue of A and w is the estimated vectorof weights.Saaty [2] uses the largest eigenvalue �max of A when solving for w in

Aw = �maxw: (5)

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2237

Saaty has shown that �max is always greater than or equal to n and that if its value is close to n,then the estimated vector of weights w solves (3) approximately. In addition, Saaty used �max todevelop a measure of the consistency of the matrix A. The consistency index (CI) is given by

CI = (�max − n)=(n− 1): (6)

The consistency ratio (CR) is given by

CR = CI=RI: (7)

The random index (RI) is the average CI of a large number of randomly generated matrices. RIdepends on the order of the matrix. A CR of 0.10 or less is considered acceptable (see [2]).The LLS method has also been used to estimate the vector of weights. In LLS, the weights wi,

for i = 1; : : : ; n, are chosen to minimize the objectiven∑

i=1

n∑

j=1

(ln aij − lnwi + lnwj)2: (8)

Given that aij = 1=aji for all i; j = 1; 2; : : : ; n, the LLS solution is quite simple: wi is given by thegeometric mean of row i (see [2]).

3. LP approach

In this section, we develop a two-stage LP approach for generating a priority vector. In the ;rststage, we formulate a linear program that provides a consistency bound for a speci;ed pairwisecomparison matrix. In the second stage, we use the consistency bound in a linear program whosesolution is a priority vector. We illustrate our approach by constructing models for a small pairwisecomparison matrix.

3.1. First stage: linear program to establish the consistency bound

Let the equation

wi=wj = aij ij; i; j = 1; 2; : : : ; n (9)

de;ne an error ij in the estimate of the relative preference aij. If the decision maker is consistent,then, after taking the natural logarithm of (9), we have ln ij = 0.Next, we de;ne the variables for our linear program. The constants are given by n= number of

rows (columns) in the square matrix A and aij =entry for row i and column j in the matrix A. Thedecision variables are given by wi =weight of element i and ij = error factor in estimating aij.We use three transformed decision variables in our model: xi= ln(wi), yij= ln( ij), and zij= |yij|.

The ;rst-stage linear program is given by the following:

Minimizen−1∑

i=1

n∑

j=i+1

zij (10)

subject to

xi − xj − yij = ln aij; i; j = 1; 2; : : : ; n; i �= j; (11)

2238 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

zij¿yij; i; j = 1; 2; : : : ; n; i¡ j; (12)

zij¿yji; i; j = 1; 2; : : : ; n; i¡ j; (13)

x1 = 0; (14)

xi − xj¿ 0; i; j = 1; 2; : : : ; n; aij ¿ 1; (15)

xi − xj¿ 0; i; j = 1; 2; : : : ; n; aik¿ ajk ; for all k;

aiq ¿ajq for some q; (16)

zij¿ 0; i; j = 1; 2; : : : ; n; (17)

xi; yij unrestricted; i; j = 1; 2; : : : ; n: (18)

We obtain constraints (11) by taking the natural logarithm of (9). In comparison matrix A, if aijis overestimated (that is, the decision maker’s judgment of entry i versus entry j is greater than thetrue value), then aji is underestimated. We then have

ij = 1= ji; i; j = 1; 2; : : : ; n (19)

or

yij =−yji; i; j = 1; 2; : : : ; n: (20)

By obtaining the greater of yij and yji, constraints (12) and (13) identify for each i and j theelement that is overestimated and the magnitude of the error. Since the solution set to constraints(11)–(13) is in;nitely large, we can arbitrarily ;x the value of any wi without loss of generality.This is done in constraint (14) by setting w1 = 1. Note that the ;nal weights can be normalized tosum to one.There are two desirable properties of a pairwise comparison matrix—element dominance (ED)

and row dominance (RD)—that we would like to model in our linear program.A solution method preserves rank weakly if aij¿ 1 implies wi¿wj. This property is known as

ED or weak rank preservation [7]. If aij is exactly equal to 1, then an argument could be madefor either wi¿wj or wj¿wi. Therefore, we modify the de;nition of weak rank preservation inthe following way: ED is preserved if aij ¿ 1 implies wi¿wj. In our ;rst-stage formulation, ED isexplicitly enforced through constraints (15). EM and LLS do not preserve ED.We point out that if the comparison matrix has cardinal inconsistency, that is, aij¿ 1, ajk¿ 1,

and aki¿ 1, then the only feasible solution is wi = wj = wk . However, such a comparison matrixwould be highly inconsistent. We see that the ED constraints in (15) have the additional bene;t ofdetecting cardinal inconsistency.A solution method preserves rank strongly if aik¿ ajk for all k implies wi¿wj. This property

is known as RD or strong rank preservation [7]. If aik = ajk for all k, then an argument could bemade for both wi¿wj and wj¿wi. Therefore, we modify the de;nition of strong rank preservationas follows: RD is preserved if aik¿ ajk for all k and aik ¿ajk for some k implies wi¿wj. In our

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2239

;rst-stage linear program, RD is explicitly enforced through constraints (16). Both EM and LLSguarantee RD (see [7]).We point out that the xi and yij decision variables in (18) are unrestricted since they are logarithms

of positive, real numbers.The objective function (10) minimizes the sum of logarithms of positive errors in natural logarithm

space. In the nontransformed space, the objective function minimizes the product of the overestimatederrors ( ij¿ 1). Therefore, the objective function minimizes the geometric mean of all errors greaterthan one. Let z∗ be the optimal objective function value of the ;rst-stage linear program. Given aperfectly consistent matrix, there is no error in the estimate and z∗ is equal to zero (since ij=1 forall i; j = 1; 2; : : : ; n or yij = 0 for all i; j = 1; 2; : : : ; n). The notion of minimizing the geometric meanof errors ;ts well with the concept of multiplicative errors in the AHP. The objective function is, insome sense, a measure of the inconsistency in the pairwise comparison matrix, that is, the greaterthe value of the objective function, the more inconsistent is the matrix.Since the objective function minimizes the sum of n(n− 1)=2 decision variables (namely, zij for

i¡ j), we de;ne the CI within the LP framework as follows:

CILP = 2z∗=n(n− 1): (6′)

CILP is the average value of zij for elements above the diagonal in the comparison matrix. Inpreliminary computational experiments, CILP and CI (see Eq. (6)) seem to be highly correlated. Wehope to explore this connection in greater detail in future work.

3.2. Second stage: linear program to generate a priority vector

When we solve the ;rst-stage linear program, the solution set consists of all priority vectors thatminimize the product of all errors ij. It is possible that there are multiple optimal solutions tothe ;rst-stage model. In the second stage, we solve a linear program that selects from this set ofalternative optima the priority vector that minimizes the maximum of errors ij. The second-stagelinear program is given by the following:

Minimize zmax (21)

subject to

n−1∑

i=1

n∑

j=i+1

zij = z∗; (22)

xi − xj − yij = ln aij; i; j = 1; 2; : : : ; n; i �= j; (23)

zij¿yij; i; j = 1; 2; : : : ; n; i¡ j; (24)

zij¿yji; i; j = 1; 2; : : : ; n; i¡ j; (25)

zmax¿ zij; i; j = 1; 2; : : : ; n; i¡ j; (26)

x1 = 0; (27)

2240 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

1 2 3

1/2 1 1

1/3 1 1

Fig. 1. 3× 3 pairwise comparison matrix.

xi − xj¿ 0; i; j = 1; 2; : : : ; n; aij ¿ 1; (28)

xi − xj¿ 0; i; j = 1; 2; : : : ; n; aik¿ ajk ; for all k;

aiq ¿ajq for some q; (29)

zij¿ 0; i; j = 1; 2; : : : ; n; (30)

xi; yij unrestricted; i; j = 1; 2; : : : ; n; (31)

zmax¿ 0: (32)

Constraint (22) ensures that only those solution vectors that are optimal in the ;rst-stage linearprogram are feasible in the second-stage model. Recall that z∗ is the optimal objective functionvalue of the ;rst-stage model. Constraints (26) ;nd zmax, the maximum value of the errors zij.The objective function (21) minimizes zmax. Constraint (32) is the nonnegativity constraint for zmax(although this constraint is redundant). All other constraints in the second-stage model are identicalto the corresponding constraints in the ;rst-stage model.

3.3. Illustrative linear programs

We apply our two-stage LP approach to the 3 × 3 pairwise comparison matrix given in Fig. 1.The decision maker needs to specify only the values in the upper triangular part of the matrix, asthe matrix is reciprocal.The ;rst-stage model for the matrix in Fig. 1 is given by the following:

Minimize z12 + z13 + z23 (33)

subject to

x1 − x2 − y12 = 0:693; (34)

x2 − x1 − y21 =−0:693; (35)

x1 − x3 − y13 = 1:099; (36)

x3 − x1 − y31 =−1:099; (37)

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2241

x2 − x3 − y23 = 0; (38)

x3 − x2 − y32 = 0; (39)

z12 − y12¿ 0; (40)

z12 − y21¿ 0; (41)

z13 − y13¿ 0; (42)

z13 − y31¿ 0; (43)

z23 − y23¿ 0; (44)

z23 − y32¿ 0; (45)

x1 − x2¿ 0; (46)

x1 − x3¿ 0; (47)

x1 − x2¿ 0; (48)

x1 − x3¿ 0; (49)

x2 − x3¿ 0; (50)

x1 = 0; (51)

zij¿ 0; xi; yij unrestricted; i; j = 1; 2; 3: (52)

Constraints (34)–(39) enforce the comparison ratios. Constraints (40)–(45) model the absolutevalues of the errors. Constraints (46) and (47) enforce ED (element 1 dominates element 2; element1 dominates element 3). Constraints (48)–(50) enforce RD (row 1 dominates row 2; row 1 dominatesrow 3; row 2 dominates row 3). Constraint (51) sets the weight of the ;rst element to 1. We pointout that constraints (48) and (49) are redundant. When we solve the ;rst-stage model using LINDO[8], we obtain z∗ = 0:406.The second-stage model for the matrix in Fig. 1 is given by the following:

Minimize zmax (53)

subject to

z12 + z13 + z23 = 0:406; (54)

x1 − x2 − y12 = 0:693; (55)

2242 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

x2 − x1 − y21 =−0:693; (56)

x1 − x3 − y13 = 1:099; (57)

x3 − x1 − y31 =−1:099; (58)

x2 − x3 − y23 = 0; (59)

x3 − x2 − y32 = 0; (60)

z12 − y12¿ 0; (61)

z12 − y21¿ 0; (62)

z13 − y13¿ 0; (63)

z13 − y31¿ 0; (64)

z23 − y23¿ 0; (65)

z23 − y32¿ 0; (66)

zmax − z12¿ 0; (67)

zmax − z13¿ 0; (68)

zmax − z23¿ 0; (69)

x1 − x2¿ 0; (70)

x1 − x3¿ 0; (71)

x1 − x2¿ 0; (72)

x1 − x3¿ 0; (73)

x2 − x3¿ 0; (74)

x1 = 0; (75)

zmax¿ 0; (76)

zij¿ 0; xi; yij unrestricted; i; j = 1; 2; 3: (77)

When we solve the second-stage model using LINDO [8], we obtain z∗max =0:135 and the priorityvector (0.55, 0.24, 0.21). This priority vector agrees with the vector generated by EM in ExpertChoice [6].

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2243

4. Advantages of the LP approach

In this section, we discuss several advantages of our LP approach including the use of dualvariables to identify inconsistencies in the decision maker’s pairwise comparisons and extensions tohandle interval judgments.

4.1. Simplicity

Our linear programs are straightforward and easy to understand and formulate. Furthermore, bothlinear programs can be solved in very little computational time using readily available software suchas LINDO [8].In fact, the two-stage linear program is no harder, computationally, than a single-stage linear pro-

gram. Once an optimal solution is obtained in the ;rst stage, the additional second-stage constraintsmay be added, and the computations can continue from the ;rst-stage extreme point solution. Thisprocess can be automated.In addition, we remark that EM requires the solution of a nonlinear program. To be fair, it is

a rather easy one (namely, Maximize � subject to Aw = �w and eTW = 1). Nonetheless, from atheoretical point of view, linear programs are easier than nonlinear programs.

4.2. Sensitivity analysis

Every linear program allows a decision maker to perform sensitivity analysis on inputs to themodel. In particular, a decision maker might be interested in answering the following questions.Which entry in the pairwise comparison matrix should be changed to reduce inconsistency? Howmuch should the entry be changed?When traditional approaches like EM and LLS are used, it is diMcult to answer these questions.

Suppose that a decision maker estimates an entry egregiously or incorrectly inputs the entry into thepairwise comparison matrix. Using EM and LLS, it would be diMcult for the decision maker to iden-tify the oOending entry by simple inspection (we point out that Expert choice [6] can automaticallylocate inconsistencies among a decision maker’s judgments and recommend revised entries).It is easy for our LP approach to answer these questions. In the ;rst-stage linear program, the

values of the dual variables at optimality provide an indication of the egregious or incorrect entries inthe pairwise comparison matrix. In LP, we know that a dual variable with value k at optimality hasthe following interpretation: if we increase the right-hand side of the dual variable’s correspondingconstraint by one unit, then the objective function value increases by k units. Since the objectivefunction value in our ;rst-stage linear program can be thought of as a measure of inconsistency(see Section 3.1), the dual variable that corresponds to each constraint in (11) measures the amountby which the inconsistency value increases if the corresponding right-hand side constant (that is,ln aij) increases by one unit. Since we are in natural logarithm space, a dual variable with value kat optimality says that the objective function value (inconsistency value) increases by k units whenthe corresponding aij increases by a factor of e, the base of the natural logarithm. This is usefulinformation for a decision maker. It is now possible to identify which aij to change in order todecrease the inconsistency value by the greatest amount. In addition, a dual variable with a negative

2244 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

1 [5,7] [2,4]

[1/7,1/5] 1 [1/3,1/2]

[1/4,1/2] [2,3] 1

Fig. 2. 3× 3 pairwise comparison matrix with lower and upper bounds [‘ij; uij] for each entry.

value at optimality indicates that aij should be increased, while a positive value indicates that aijshould be decreased.We now consider the dual variables associated with the ED constraints (15). Speci;cally, when

the value of the dual variable at optimality is greater than zero for an ED constraint, the constraint isbinding. This indicates that the pairwise comparison entry corresponding to that particular constraintmight be Pawed. This set of dual variables can help a decision maker detect cardinal inconsistency ina pairwise comparison matrix (detecting cardinal inconsistency is not possible using EM and LLS).We illustrate this capability of our LP approach in Section 5.

4.3. Modeling interval judgments

In a pairwise comparison matrix, typically aij is a single number that estimates wi=wj. Supposethat, instead of a single number, an interval is speci;ed with a lower bound ‘ij and an upper bounduij on the estimate. For example, consider the 3 × 3 pairwise comparison matrix given in Fig. 2.We see that entry a12 is a number between 5 and 7. Since the matrix is reciprocal, a21 is a numberbetween 1

7 and15 .

Arbel and Vargas [9] treat the interval bounds as hard constraints. They develop two techniquesto generate priority vectors when interval judgments are used: preference simulation and preferenceprogramming. In preference simulation, several comparison matrices are obtained by sampling fromthe speci;ed intervals. The EM approach is then applied to each matrix to produce a priority vector.The average of the feasible priority vectors gives the ;nal set of weights. Of course, this approachcan be extremely ineMcient and computationally burdensome when most of the priority vectors areinfeasible. This can happen as a consequence of several tight interval judgments.Preference programming uses linear inequalities and equations of the form

‘ij6wi=wj6 uij; i; j = 1; 2; : : : ; n; i¡ j; (78)

n∑

i=1

wi = 1; (79)

wi¿ 0; i = 1; 2; : : : ; n; (80)

where ‘ij and uij are the lower and upper bounds of the speci;ed interval. If a solution to this set ofequations exists, it de;nes an n-dimensional priority space. The arithmetic mean of the vertices ofthis feasible region becomes the ;nal priority vector. No attempt is made to identify the best vectorin the feasible region.The ;rst-stage linear program speci;ed by (10)–(18) can be revised to handle the interval AHP

problem in the following way. Each entry aij is the geometric mean of the interval bounds, that is,

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2245

aij = (‘ij × uij)1=2. We use the geometric mean in order to preserve the inverse reciprocal propertyof the matrix: (‘ij × uij)1=2 = 1=((1=‘ij)× (1=uij))1=2.In the ;rst-stage linear program, we replace constraints (15) and (16) with the following con-

straints:

xi − xj¿ ln ‘ij; i; j = 1; 2; : : : ; n; i¡ j; (81)

xi − xj6 ln uij; i; j = 1; 2; : : : ; n; i¡ j: (82)

If ‘ij ¿ 1, then the priority vector is bound by this value and generates weights such that wi ¿wj.Thus, when ‘ij ¿ 1, a constraint in (81) behaves like an element dominance constraint for aij ¿ 1.Similarly, when uij ¡ 1, a constraint in (82) behaves like an ED constraint for aij ¡ 1. The ;rst-stagemodel for handling interval judgments has objective function (10) and constraints (11)–(14), (81),(82), (17), and (18).The ;rst-stage linear program that models the interval judgments shown in Fig. 2 is given by the

following:

Minimize z12 + z13 + z23 (83)

subject to

x1 − x2 − y12 = 1:778; (84)

x2 − x1 − y21 =−1:778; (85)

x1 − x3 − y13 = 1:040; (86)

x3 − x1 − y31 =−1:040; (87)

x2 − x3 − y23 =−0:896; (88)

x3 − x2 − y32 = 0:896; (89)

z12 − y12¿ 0; (90)

z12 − y21¿ 0; (91)

z13 − y13¿ 0; (92)

z13 − y31¿ 0; (93)

z23 − y23¿ 0; (94)

z23 − y32¿ 0; (95)

x1 − x2¿ 1:609; (96)

2246 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

1 [8,9] 2

[1/9,1/8] 1 [1/7,1/5]

1/2 [5,7] 1

Fig. 3. 3× 3 mixed pairwise comparison matrix.

x1 − x26 1:946; (97)

x1 − x3¿ 0:693; (98)

x1 − x36 1:386; (99)

x2 − x3¿− 1:099; (100)

x2 − x36− 0:693; (101)

x1 = 0; (102)

zij¿ 0; xi; yij unrestricted; i; j = 1; 2; 3: (103)

Constraints (84)–(89) enforce the comparison ratios. Constraints (90)–(95) model the absolutevalues of the errors. Constraints (96)–(101) enforce the bounds on the ratios of the weights. Con-straint (102) sets the weight of the ;rst element to 1. We do not formulate the second-stage linearprogram due to space considerations.

4.4. Mixed pairwise comparison matrices

Our LP approach can easily handle a mixture of single entries and interval bounds in a pairwisecomparison matrix. That is, some entries are single numbers aij and some entries have intervalbounds of the form [‘ij; uij].The ;rst-stage linear program that models the mixed pairwise comparison matrix shown in Fig. 3

is given by the following:

Minimize z12 + z13 + z23 (104)

subject to

x1 − x2 − y12 = 2:138; (105)

x2 − x1 − y21 =−2:138; (106)

x1 − x3 − y13 = 0:693; (107)

x3 − x1 − y31 =−0:693; (108)

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2247

x2 − x3 − y23 =−1:778; (109)

x3 − x2 − y32 = 1:778; (110)

z12 − y12¿ 0; (111)

z12 − y21¿ 0; (112)

z13 − y13¿ 0; (113)

z13 − y31¿ 0; (114)

z23 − y23¿ 0; (115)

z23 − y32¿ 0; (116)

x1 − x2¿ 2:079; (117)

x1 − x26 2:197; (118)

x2 − x3¿− 1:946; (119)

x2 − x36− 1:609; (120)

x1 − x3¿ 0; (121)

x1 = 0; (122)

zij¿ 0; xi; yij unrestricted; i; j = 1; 2; 3: (123)

Constraints (105)–(110) enforce the comparison ratios. Constraints (111)–(116) model the abso-lute values of the errors. Constraints (117)–(120) enforce the bounds on the ratios of the weights.Constraint (121) enforces ED (there are no RD constraints). Constraint (122) sets the weight of the;rst element to 1. We do not formulate the second-stage linear program due to space considerations.

4.5. Modeling group decisions

An important application of AHP involves group decision making (e.g., [10]). Suppose there aren decision makers. The most common approach used in AHP is for each decision maker to ;ll in acomparison matrix independently such that akij denotes the comparison of element i to element j fordecision maker k (k = 1; 2; : : : ; n). The individual judgments of the n decision makers are combinedusing the geometric mean to produce entries

aij = [a1ij × a2ij × · · · × anij]1=nwhich collectively determine an overall comparison matrix A. EM is applied to A to obtain thepriority vector.

2248 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

An alternative direction is to take advantage of the LP approach to mixed pairwise comparisonmatrices, as discussed in Section 4.4. Instead of computing the geometric mean aij, we can computeinterval bounds [‘ij; uij] where

‘ij =min{a1ij ; a2ij ; : : : ; anij} and

uij =max{a1ij ; a2ij ; : : : ; anij} for i¡ j:

If ‘ij = uij, we use a single number, rather than an interval, in order to avoid over-constraining thelinear program. This approach is extremely Pexible. For example, if there are many decision makers(n is large), one can eliminate the high and low values (i.e., eliminate outliers) and compute intervalbounds [‘ij; uij] or a single number from the remaining n− 2 values.

4.6. Other advantages

The ability to ensure ED and RD via LP constraints is a major advantage oOered by the LPapproach. In addition, one can think of ED as a mechanism for providing limited protection againstrank reversal. If aij ¿ 1, then, irrespective of the number of alternatives added or deleted, wi¿wjas a result of the ED constraints.

5. Computational experiment: #ve pairwise comparison matrices

In this section, we formulate our ;rst- and second-stage linear programs, solve the models, anddiscuss results for ;ve square, reciprocal, pairwise comparison matrices that range in size from fourto seven rows.

5.1. Matrix 1

In Fig. 4, we give Matrix 1: a 7× 7 pairwise comparison matrix that exhibits cardinal inconsis-tency. That is, for three elements i; j, and k; aij ¿ 1, ajk ¿ 1, and aki ¿ 1. The weights generated byEM, LLS, and a second-stage linear program that contains both ED and RD constraints are shownin Table 1. We point out that this matrix has a consistency ratio of 0.10. Recall that both EM andLLS guarantee RD.A close inspection of Matrix 1 reveals that the key sources of inconsistency are the judgments

involving elements 4, 6, and 7. We see that element 4 is less important than element 6 (a46 = 12),

1 5 1 4 2 6 71/5 1 1/8 1 1/3 4 21 8 1 5 3 3 3

1/4 1 1/5 1 1/2 1/2 21/2 3 1/3 2 1 7 21/6 1/4 1/3 2 1/7 1 1/21/7 1/2 1/3 1/2 1/2 2 1

Fig. 4. Matrix 1.

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2249

Table 1Priority vectors for Matrix 1

Weight EM LLS Second-stage LP model

RD RD ED and RD

w1 0.291 0.312 0.303w2 0.078 0.073 0.061w3 0.300 0.293 0.303w4 0.064 0.064 0.061w5 0.159 0.157 0.152w6 0.051 0.044 0.061w7 0.058 0.057 0.061

ED: Element dominance. RD: Row dominance.

- 0 - 0 0 0 0- - - - - 0 0- 0 - 0 0 0 0- - - - - - 0- 0 - 0 - 0 0- - - 4 - - -- - - - - 0 -

Fig. 5. Optimal values of the dual variables (;rst-stage linear program with ED and RD constraints) for Matrix 1corresponding to each element dominance constraint (shown only for aij ¿ 0).

element 6 is less important than element 7 (a67 = 12), and element 7 is less important than element

4 (a74 = 12). Given this instance of cardinal inconsistency, the optimal solution to the second-stage

model with ED constraints and RD constraints has w4 = w6 = w7. In general, when an ED con-straint is binding, the two weights corresponding to that constraint are equal at optimality in theLP model.It is diMcult to detect cardinal inconsistency in a matrix by simple inspection alone. However,

we can detect inconsistency by examining the optimal values of the dual variables correspondingto the ED constraints given in (15). In Fig. 5, we show the optimal values of the dual variablescorresponding to the ED constraints in the ;rst-stage model with ED and RD constraints for Matrix1. We see that only the ED constraint corresponding to a64 is binding (the optimal value of the dualvariable is positive). Hence, we suspect that a64 = 2 and a46 = 1

2 are incorrect judgments. Expertchoice can also be used to identify sources of cardinal inconsistency. The software identi;es a26 asthe “most inconsistent judgment.”Furthermore, in Fig. 6, we show the optimal values of the dual variables corresponding to the

weight constraints in (11) for the ;rst-stage model with ED and RD constraints of Matrix 1. Anonzero value indicates that an entry in the pairwise comparison matrix might be increased in orderto reduce inconsistency. The optimal values of the dual variables corresponding to a46 and a67(among others) are nonzero. Thus, we can reduce inconsistency by increasing the value of eitherentry in Matrix 1.

2250 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

0 0 0 -2 -2 -2 0- 0 -2 0 -2 0 0- - 0 0 0 -2 -2- - - 0 0 -2 0- - - - 0 0 -2- - - - - 0 -2- - - - - - 0

Fig. 6. Optimal values of the dual variables (;rst-stage linear program with ED and RD constraints) for Matrix 1corresponding to each aij constraint. Lower triangle values are the negative of the upper triangle values.

1 2 2.5 8 51/2 1 1/1.5 7 5

1/2.5 1.5 1 5 31/8 1/7 1/5 1 1/21/5 1/5 1/3 2 1

Fig. 7. Matrix 2.

0 0 0 -2 -2- 0 -2 0 0- - 0 -2 0- - - 0 -2- - - - 0

Fig. 8. Optimal values of the dual variables (;rst-stage linear program with ED and RD constraints) for Matrix 2corresponding to each aij constraint. Lower triangle values are the negative of the upper triangle values.

We point out that, for the ;rst-stage model with ED and RD constraints of Matrix 1, the optimalvalues of the dual variables corresponding to the RD constraints in (16) are all zero. In this case,none of these constraints is binding.

5.2. Matrix 2

Our LP approach has the ability to model the situation in which a decision maker compares twoalternatives i and j directly, and wants to enforce that alternative i is more important than alternativej in the ;nal ranking. Our LP approach can preserve this ordering through ED constraints. Forexample, suppose a father and daughter are using AHP to help decide on the right college for herto attend. The father may want to ensure that “cost” is more important than any factor other than“academic quality.”In Fig. 7, we give Matrix 2: a 5×5 pairwise comparison matrix for which the decision maker has

speci;ed that w26w3. The weights generated by EM, LLS, and our second-stage linear programare presented in Table 2. This matrix has a consistency ratio of 0.03. We observe that EM and LLSviolate the ED constraint since w2¿w3.In Fig. 8, we show the optimal values of the dual variables corresponding to the weight constraints

in (11) for the ;rst-stage model with ED and RD constraints of Matrix 2. We observe that the mainsources of inconsistency in this matrix come from entries a14, a15, a23, a34, and a45. Thus, the

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2251

Table 2Priority vectors for Matrix 2

Weight EM LLS Second-stage LP model

RD RD ED and RD

w1 0.419 0.422 0.441w2 0.242 0.239 0.221w3 0.229 0.227 0.221w4 0.041 0.041 0.044w5 0.070 0.071 0.074

ED: Element dominance. RD: Row dominance.

1 [2,5] [2,4] [1,3][1/5,1/2] 1 [1,3] [1,2][1/4,1/2] [1/3,1] 1 [1/2,1][1/3,1] [1/2,1] [1,2] 1

Fig. 9. Matrix 3.

Table 3Priority vectors for Matrix 3

Preference simulationa Preference programminga Second-stage LP model

Weight Minimum Average Maximum Standard deviation

w1 0.369 0.470 0.552 0.037 0.469 0.425w2 0.150 0.214 0.290 0.026 0.201 0.212w3 0.093 0.132 0.189 0.016 0.146 0.150w4 0.133 0.184 0.260 0.023 0.185 0.212

aResults from Arbel and Vargas [9].

inconsistency of Matrix 2 can be reduced by increasing the value of any one of these ;ve aijentries.

5.3. Matrix 3

In Fig. 9, we give Matrix 3: a 4 × 4 pairwise comparison matrix with upper and lower bounds[‘ij; uij] speci;ed for each entry. In Table 3, we present weights that were generated by preferencesimulation and preference programming (these results are due to Arbel and Vargas [9]) and by oursecond-stage model. In preference simulation, the feasible priority vectors are averaged to give the;nal set of weights (this is denoted by Average in Table 3; we also show the minimum value andmaximum value for each wi from the feasible priority vectors that were generated by the simulation).

2252 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

1 [2,4] 4 [4.5,7.5] 1[1/4,1/2] 1 1 2 [1/5,1/3]

1/4 1 1 [1,2] 1/2[1/7.5,1/4.5] 1/2 [1/2,1] 1 1/3

1 [3,5] 2 3 1

Fig. 10. Matrix 4.

Table 4Priority vectors for Matrix 4

Weight EM Second-stage LP model

w1 0.377 0.413w2 0.117 0.103w3 0.116 0.103w4 0.076 0.071w5 0.314 0.310

For our ;rst-stage model, the optimal values of the dual variables corresponding to the lower andupper bound constraints in (81) and (82) are all zero. However, if some of the values of the dualvariables were nonzero, this would enable the decision maker to determine which of the bounds aretoo tight and which of the bounds could be changed in order to reduce inconsistency in the matrix.

5.4. Matrix 4

In Fig. 10, we give Matrix 4: a 5 × 5 pairwise comparison matrix that has a mixture of singleaij entries and interval entries where upper and lower bounds [‘ij; uij] are speci;ed. In Table 4, wepresent weights that were generated by EM and our second-stage model with ED constraints forevery aij ¿ 1 entry in the matrix. Of course, EM was not designed to handle an interval entry, sothat it was necessary to convert every interval entry into a single aij entry. We accomplished theconversion by computing the geometric mean of every lower bound and upper bound and then usedthe geometric mean as the single entry in the matrix. We observe that the weights generated by EMviolate one of the four interval constraints (the interval [1=5; 1=3] is violated).

5.5. Group AHP example

To illustrate the material in Section 4.5, we conducted an experiment. Four decision makers(graduate students) were given ;ve geometric ;gures (taken from Gass [11, Chapter 24]) and wereasked to compare (by visual inspection) the area of ;gure i to the area of ;gure j for i¡ j. Lowerbounds and upper bounds were determined as speci;ed in Section 4.5. The lower bounds are providedin Fig. 11 and the upper bounds are shown in Fig. 12. Since ‘34 = u34 = 4:0000 (indicated in bold),we use a single number for a34, rather than an interval.We now seek to compare the results generated by EM, LLS, and the LP model. To apply EM

and LLS, we must ;rst compute the geometric means

aij = [a1ij × a2ij × a3ij × a4ij]1=4:

B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254 2253

1.0000 2.0000 1.5000 4.5000 0.50000.2500 1.0000 0.5000 2.0000 0.12500.5000 1.3333 1.0000 4.0000 0.25000.1250 0.3333 0.2500 1.0000 0.06251.3333 2.5000 1.5000 6.0000 1.0000

Fig. 11. Lower bounds from the group.

1.0000 4.0000 2.0000 8.0000 0.75000.5000 1.0000 0.7500 3.0000 0.40000.6667 2.0000 1.0000 4.0000 0.66670.2222 0.5000 0.2500 1.0000 0.16672.0000 8.0000 4.0000 16.0000 1.0000

Fig. 12. Upper bounds from the group.

1.0000 2.7832 1.6119 6.7007 0.61480.3593 1.0000 0.5533 2.2134 0.22960.6204 1.8072 1.0000 4.0000 0.48890.1492 0.4518 0.2500 1.0000 0.09911.6266 4.3559 2.0453 10.0908 1.0000

Fig. 13. Geometric mean of the group.

Table 5Priority vectors for geometry experiment

Weight EM LLS Second-stage LP model Actual geometric areas

w1 0.272 0.272 0.277 0.273w2 0.096 0.096 0.095 0.091w3 0.178 0.178 0.172 0.182w4 0.042 0.042 0.041 0.045w5 0.412 0.412 0.414 0.409

The results are displayed in Fig. 13. The LP model uses interval bounds and one single number.The three priority vectors and the actual geometric areas (normalized to sum to one) are presentedin Table 5. They are remarkably similar.

6. Conclusions

In this paper, we have presented an intuitively appealing LP approach for estimating priorityvectors in the AHP. Our LP approach has several advantages over more traditional approaches. Oneadvantage is that users are more likely to understand output from an LP model than know abouteigenvectors or LSS. A second involves sensitivity analysis. Our measure of inconsistency has amore intuitive interpretation than that in EM. A third is that the LP approach can model pairwise

2254 B. Chandran et al. / Computers & Operations Research 32 (2005) 2235–2254

comparison matrices that have single number entries, interval entries, or a mixture of both types ofentries (to our knowledge, we are the ;rst to consider matrices of this type). We have demonstratedan extension of this approach to AHP-based group decision making. Finally, the new approachensures ED and RD via LP constraints.We point out that this paper is not an attempt to resolve the debate as to which is the correct

approach to use in deriving priority vectors. Instead, we simply present an alternative approach thathas some interesting and desirable properties.Also, we do not claim that adding ED and RD constraints is always the right thing to do,

though it certainly seems reasonable. If the decision maker speci;cally indicates a preference forone alternative over another (recall the college selection example in Section 5.2), a method thatproduces a solution with the weights reversed might not be very helpful or insightful. By includingthese constraints and obtaining the optimal values of the corresponding dual variables, the decisionmaker can easily identify inconsistencies in the pairwise comparison matrix. Ultimately, the choice ofwhich constraints to include or omit depends on the importance of rank preservation to the decisionmaker.Finally, we formulated LP models for several pairwise comparison matrices and compared the LP

results to those produced by two widely used methods. In general, the weights generated by ourlinear programs are similar to those produced by EM and LLS.In addition, we observed that the rankings of alternatives are nearly the same for all three methods.

Our LP approach is easy to implement and is computationally inexpensive.

Acknowledgements

The authors thank Larry Bodin for reading a draft of this paper and providing helpful commentsand suggestions.

References

[1] Saaty T. A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology 1977;15:234–81.

[2] Saaty T. The analytic hierarchy process. New York: McGraw-Hill; 1980.[3] Wasil E, Golden B. Celebrating 25 years of AHP-based decision making. Computers and Operations Research

2003;30:1419–20.[4] Golden B, Wasil E, Harker P. The analytic hierarchy process: applications and studies. Berlin: Springer; 1989.[5] Harker P, Vargas L. The theory of ratio scale estimation: Saaty’s analytic hierarchy process. Management Science

1987;33:1383–403.[6] Expert choice. Arlington, VA: Expert Choice; 2003.[7] Saaty T, Vargas L. Inconsistency and rank preservation. Journal of Mathematical Psychology 1984;28:205–14.[8] LINDO. Chicago, IL: LINDO Systems; 2003.[9] Arbel A, Vargas L. Preference simulation and preference programming: robustness issues in priority derivation.

European Journal of Operational Research 1993;69:200–9.[10] Condon E, Golden B, Wasil E. Visualizing group decisions in the analytic hierarchy process. Computers and

Operations Research 2003;30:1435–45.[11] Gass S. Decision making, models and algorithms: a ;rst course. New York: Wiley; 1985.