A GP-AHP method for solving group decision-making fuzzy AHP ...

Computers & Operations Research 29 (2002) 1969–2001 

www.elsevier.com/locate/dsw 

AGP-AHP method for solving group decision-making fuzzy 

AHP problems 

Abstract 

Chian-Son Yu ∗ 

Department of Information Management, Shih Chien University, Taipei 10497, Taiwan, ROC 

Received 1 September 2000; received in revised form 1 May 2001 

The analytic hierarchy process (AHP) elicits a corresponding priority vector interpreting the preferred 

information from the decision-maker(s), based on the pairwise comparison values of a set of objects. 

Since pairwise comparison values are the judgments obtained from an appropriate semantic scale, in 

practice the decision-maker(s) usually give some or all pair-to-pair comparison values with an uncertainty 

degree rather than precise ratings. By employing the property of goal programming (GP) to treat a 

fuzzy AHP problem, this paper incorporates an absolute term linearization technique and a fuzzy rating 

expression into a GP-AHP model for solving group decision-making fuzzy AHP problems. In contrast to 

current fuzzy AHP methods, the GP-AHP method developed herein can concurrently tackle the pairwise 

comparison involving triangular, general concave and concave–convex mixed fuzzy estimates under a 

group decision-making environment. 

Scope and purpose 

Many real world decision problems involve multiple criteria in qualitative domains. As expected, such 

problems will be increasingly modeled as multiple criteria decision-making problems, which involve 

scoring on subjective=qualitative domains. This results in a class of signi cant problems for which an 

evaluation framework, which handles occurrences of seeming intransitivity and inconsistency will be 

required. Another interesting issue of group decision-making analysis is how to deal with disagreements 

between two or more di erent rankings within an alternative set. These phenomena are likely to appear 

in qualitative=subjective domains where the decision-making environment is ambiguous and vague. 

Therefore, this study proposes a GP-AHP model that is su ciently robust to permit con ict and imprecision. 

Numerical examples demonstrate the e ectiveness and applicability of the proposed models in 

deriving the most promising priority vector from a fuzzy AHP problem within a group decision-making 

environment. ? 2002 Elsevier Science Ltd. All rights reserved. 

Keywords: Group decision-making; Fuzzy AHP; Fuzzy numbers 

∗ Tel.: +886-2-25381111; fax: +886-2-25336293. 

E-mail address: csyu@mail.scc.edu.tw (C.-S. Yu). 

0305-0548/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. 

PII: S 0305-0548(01)00068-5

1970 C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 

1. Introduction 

The analytic hierarchy process (AHP) pioneered in 1971 by Saaty [1] is a widespread 

decision-making analysis tool for modeling unstructured problems in areas such as political, 

economic, social, and management sciences. Based on the pair-by-pair comparison values for 

a set of objects, AHP is applied to elicit a corresponding priority vector that represents preferences. 

Since pairwise comparison values are the judgments obtained using a suitable semantic 

scale, it is unrealistic to expect that the decision-maker(s) have either complete information or 

a full understanding of all aspects of the problem [2–10]. Many researchers [9,11–13] have also 

noted that fuzzyness and vagueness are characteristics of many decision-making problems. It has 

been inferred that good decision-making models and decision-makers must tolerate vagueness 

or ambiguity and be able to function in such situations. 

Since it is di cult to map qualitative preferences to point estimates, a degree of uncertainty 

will be associated with some or all pairwise comparison values in an AHP problem. The problem 

of generating such a priority vector in the uncertain pair-to-pair comparison environment is called 

the fuzzy AHP problem, as described below. 

Assume there is a set of objects {1; 2;:::;n}. The evaluator’s preference regarding these 

objects is represented numerically via a positive reciprocal matrix A = {ãij} with ãij =1=ãji, 

where ãij is a fuzzy number that is the numerical equivalent of the comparison between objects 

‘i’ and ‘j’. For the comparison of each pair of objects ‘i’ and ‘j’, the ultimate objective is to 

generate the most promising priority vector w =(w1;w2;:::;wn) such that the ratio wij = wi=wj 

is the best approximation to the evaluator’s speci ed ãij. Therefore, a fuzzy AHP problem can 

be formulated as follows: 

Problem 1. 

Minimize � 

� 

� � 

� 

� 

i 

j¿i 

wi 

− ãij 

wj 

� 

� 

� 

� 

Subject to wi; ãij¿0 (i; j) ∈ (I; J )={(i; j) | 1 6 i¡j6 n}; � 

wi =1; 

where ãij represents the fuzzy number to which object i is preferred over object j; n is the 

number of objects, the ratio wi=wj is the comparison between each pair of objects ‘i’ and ‘j’, 

and ãij =1=ãji from a fuzzy positive reciprocal matrix. 

2. Literature review 

Afuzzy AHP problem was rst presented in 1980 by Graan [14]. His method of extracting 

the priority vector from an AHP problem with fuzzy ratings falls into two levels. The 

rst level assigns fuzzy weights ˜i to each criteria Ci (i =1; 2;:::;m) while the second level 

assigns fuzzy weights ˜ ij (j =1; 2;:::;n) to the alternatives Aj under each criterion Ci sepa- 

rately. Then the nal fuzzy priority vector ˜W =(˜w1; ˜w2;:::; ˜wn) can be obtained by calculating 

;j=1; 2;:::;n. 

� m 

i=1 ˜i ˜ ij 

i

C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 1971 

Laarhoven et al. [8] in 1983 proposed the following mathematical model using logarithmic 

least squares method (LLSM) to solve Problem 1: 

Minimize � ij � 

(ln wi − ln wj − ln aijk) 2 

i¡j k=1 

⎛ 

Subject to ‘i ⎝ 

mi 

ui 

⎛ 

⎝ 

⎛ 

⎝ 

n� 

j=1;j�=i 

n� 

j=1;j�=i 

n� 

j=1;j�=i 

ij 

ij 

ij 

⎞ 

⎠ − 

⎞ 

⎠ − 

⎞ 

⎠ − 

n� 

j=1;j�=i 

n� 

j=1;j�=i 

n� 

j=1;j�=i 

ijuj = 

ijmj = 

ij‘j = 

n� 

ij � 

j=1;j�=i k=1 

n� 

ij � 

j=1;j�=i k=1 

n� 

ij � 

j=1;j�=i k=1 

‘ijk; i=1; 2;:::;n; 

mijk; i=1; 2;:::;n; 

uijk; i=1; 2;:::;n; 

where aijk (k =1; 2;:::; ij) are ij estimates for wi=wj ( ij can equal zero, if no comparisons 

are available; equal to or greater than one, in which case there are multiple comparisons) and 

‘ijk, mijk, and uijk are lower, modal, and upper values of ln ãijk = − ln ãjik, respectively. LLSM 

is applied to reduce the fractional form wi=wj in Problem 1 into linear form ln wi − ln wj. The 

quadratic form (ln wi − ln wj − log aijk) 2 is employed to ensure the variance from |wi=wj − ãij| 

in Problem 1 ¿ 0. 

Boender et al. [2] in 1989 improved Laarhoven et al.’s model as follows: 

m� m� � 

Minimize 

{ln(wijk‘) − ln(ai‘)+ln(aju)} 2 + {ln(wijkm) − ln(aim)+ ln(ajm)} 2 

Subject to ln(ai‘) 

i=1 j=i+1 k∈Dij 

+ {ln(wijku) − ln(aiu)+ln(aj‘)} 2 

ln(aim) 

ln(aiu) 

m� 

j�=i 

m� 

j�=i 

m� 

ij − 

ij − 

ij − 

m� 

j�=i 

m� 

j�=i 

m� 

ij ln(aju)= 

ij ln(ajm)= 

ij ln(aj‘)= 

m� � 

ln(wijk‘); i=1; 2;:::;m; 

j�=i k∈Dij 

m� � 

ln(wijkm); i=1; 2;:::;m; 


m� � 

ln(wijku); i=1; 2;:::m; 

j�=i j�=i 


where wijk‘; wijkm; wijku are derived lower, modal, and upper priority vectors, respectively. Fuzzy 

ratings ãij are represented by ln(ai‘); ln(aim); ln(aiu); ln(aj‘), ln(ajm) and ln(aju). 

Over the last two decades, numerous methods [2–9,11,12,15–22] have been developed to 

generate the priority vector by comparing all pairs of criteria and decision alternatives under 

a fuzzy environment. Despite these e orts, conventional fuzzy AHP methods continue to use 

either repetitive extension principal procedures like -cut approaches to optimize the intersection 

among membership functions or tedious arithmetic calculations like geometric mean techniques


to deal with the operation among fuzzy values. The major shortcoming of conventional fuzzy 

AHP methods is not the requirement of repeated extension principal processes or tedious arithmetic 

computations, but rather in ignoring the interactions among the fuzzy variables and the 

decision-makers. Furthermore, when conventional methods are employed, only triangular membership 

functions can be interpreted while more general concave, convex, or concave–convex 

mixed membership functions cannot be conveniently interpreted. Another disadvantage of the 

traditional fuzzy AHP method is the derived solution is a fuzzy priority vector where extra 

defuzzi cation calculation is required to produce a crisp solution. 

3. Problem formulations 

Goal programming (GP) methods employed to derive the priority vector from the reciprocal 

matrix of pairwise comparison has been reported [23–28]. The primary merit of combining 

the GP and LLSM techniques to formulating an AHP problem as a GP-AHP model is that 

this model can easily to treat an AHP problem with fuzzy, point, and interval ratings as well 

as some ratings without available information simultaneously. Furthermore, in a deterministic 

AHP problem using GP for priority derivation cannot only ful ll the four axioms that Fichtner 

[26] proposed, but also ful lls another axiom called single outlier neutralization [23,24]. Hence, 

within fuzzy AHP problems, this study employs GP and LLSM properties to minimize the 

variance from vague pairwise comparisons. Accordingly, Problem 1 can be reformulated as 

follows: 

Problem 2. 

Minimize � � 

| (log vi − log vj) − log ãij| 

i 

j¿i 

Subject to vi; ãij ¿ 0; (i; j) ∈ (I; J )={(i; j) | 1 6 i¡j6 n}; 

where ãij represents fuzzy numbers, the un-normalized vector V =(v1;:::;vn) will be normalized 

to produce the vector W =(w1;:::;wn) with vi=vj = wi=wj, and � 

i wi =1. 

In many practical applications, an AHP problem frequently encounters not only imprecise 

ratings, but also a group evaluation [10,21,29–33] such as that from a committee or expert 

representatives. Hence, this study incorporates a new piecewise linear expression and an absolute 

term linearization means into a GP-AHP model to solve group decision-making fuzzy AHP 

problems. Treating group decision-making fuzzy AHP problems with triangular membership 

functions, which are based on Corollary 1 (will be discussed in Section 4), Problem 2 can then 

be programmed as follows: 

Problem 3. 

Minimize 

n� 

n� 

i=1 j¿i e=1 

E� 

| (log vi − log vj) − log ae ij| (3.1)

Maximize 


n� 

n� 

E� 

i=1 j¿i e=1 

(log a e ij) 

Subject to log a e ij = { (log a e ij)+(s e ij;R − s e ij;L) log a e ij;2 

−(s e ij;R − s e ij;L)d e ij + s e ij;L log a e ij;1 }=se ij;R; 

log ae ij + log ae ij;2 + de ij ¿ 0; vi;de ij;ae ij ¿ 0; 

(i; j) ∈{(i; j) | 1 6 i¡j6 n}; 

(3.2) 

where ae ij represents the eth decision-maker’s uncertainty assessment to which object i is 

preferred more than object j; n depicts the number of objects, E represents the number of 

decision-makers, fuzzy values log ae ij are conducted by (log aeij ), the term |log vi −log vj −log ae ij | 

in (3.1) is the sum of deviations where decision-makers would like to minimize, and the term 

(log ae ij ) in (3.2) is the sum of all membership functions’ grades where decision-makers want 

to maximize. 

To integrate (3.1) and (3.2), the following theorem is introduced. 

Theorem 1. Problem 3 is equivalent to the following problem: 

Problem 4. 

Minimize 

n� 

n� 

i=1 j¿i e=1 

E� 

| (log vi − log vj) − log ae ij |− 

n� 

n� 

E� 

i=1 j¿i e=1 

(log a e ij) 

Subject to log a e ij = { (log a e ij)+(s e ij;R − s e ij;L) log a e ij;2 − (se ij;R − s e ij;L)d e ij 

+ s e ij;Llog a e ij;1 }=se ij;R; 

log ae ij + log ae ij;2 + de ij ¿ 0; 

vi;de ij;ae ij ¿ 0; (i; j) ∈{(i; j) | 1 6 i¡j6 n}; 

(3.3) 

where is a trade-o weight speci ed by =(n×E)=|M 1 −M 0 | for n×E membership functions. 

Proof. The lower ratio deviation within |log vi − log vj − log ae ij | in (3.1) implies that the higher 

promising priority vector vij = vi=vj will be generated. The higher grade of membership functions 

(log ae ij ) in (3.2) implies that the higher evaluators’ desirability will be derived. In solving 

problems with multiple and non-commensurable goals a single solution capable of optimizing 

all the goals generally does not exit. Consequently, in certain sense the optimal solution in 

Problem 3 is a trade-o solution. 

The optimization of simultaneously considering both an objective function and a group of 

membership functions constitutes a trade-o problem. By assuming each (log aij) equals one 

and after solving Problem 3, the obtained solution is denoted by M 1 . In contrast, when each 

(log aij) equal zero, after Problem 3 is solved M 0 denotes the obtained solution. The technique 

for order preference by similarity to ideal solution (TOPSIS) initially developed by Yoon et al.


Fig. 1. Atriangle membership function. 

[34] and LINnear programming techniques for multidimensional analysis of preference (LIN- 

MAP) initially presented by Srinivasan et al. [34] are adopted here. Nevertheless, by computing 

=(n × E)=|M 1 − M 0 | for n × E membership functions, the trade-o weighting value, ,is 

determined. This proves the theorem. 

4. Treating fuzzy numbers 

To interpret fuzzy numbers, an obvious representation of a triangular membership function 

without symmetric restriction is rst introduced. 

Proposition 1. Fig. 1 depicts (log aij) as a triangle membership function of a fuzzy number 

log aij; where log aij;k (k =1; 2; 3) are the possible lowest; middle and highest values; respectively. 

Let 

sij;L = (log aij;2) − (log aij;1) 

log aij;2 − log aij;1 

represent the slope of the line segment between log aij;1 and log aij;2; and 

sij;R = (log aij;3) − (log aij;2) 

aij;3 − aij;2 

represent the slope of the line segment between log aij;2 and log aij;3. Hence (log aij) can then 

be represented as 

(log aij)=sij;L(log aij − log aij;1)+ sij;R − sij;L 

(|log aij − log aij;2| + log aij − log aij;2); 

2 

(4.1) 

where |o| is the absolute value of o.


Fig. 2. Afuzzy value log a12. 

Proof. This proposition can be veri ed as follows: 

(i) If log aij 6 log aij;2 then 


(|log aij − log aij;2| + log aij − log aij;2) 

2 

= sij;L(log aij − log aij;1): 

(ii) If log aij;2 6 log aij 6 log aij;3 then 



2 

= sij;L(log aij − log aij;1)+(sij;R − sij;L)(log aij − log aij;2) 

= sij;L(log aij;2 − log aij;1)+sij;R(log aij − log aij;2): 

This proposition is then proved. Consider the following example. 

Example 1. 

Maximize (log a12) 

Subject to a12 ¿ 0; 

where log a12 is a fuzzy value depicted in Fig. 2. 

Through Proposition 2, (log a12) can be expressed as follows: 

−14:93718 − 5:67887 

(log a12)=5:67887(log a12 − log 2) + 

2 

(|log a12 − log 3| + log a12 − log 3)


=5:67887(log a12 − log 2) − (20:61605=2) 

(|log a12 − log 3| + log a12 − log 3): (4.2) 

To linearize the absolute term, the following proposition is presented: 

Proposition 2. An absolute term program with a negative coe cient is expressed as follows: 

{Maximize Z = c=2 (|f(X ) − g| + f(X ) − g) subject to X ∈ F (F is a feasible set); c is a 

negative coe cient (i:e:; c 6 0) and g is a given constant} can be transformed into following 

linear program 

{Maximize ZZ = c(f(X ) − g + d) subject to f(X ) − g + d ¿ 0;d¿ 0;c6 0;X ∈ F}. 

Proof. This proposition can be examined as follows: 

(i) Case 1: If f(x) − g ¿ 0; z= c(f(X ) − g). 

At the optimal solution d will be forced into d = 0, resulting in ZZ = c(f(X ) − g)=Z. 

(ii) Case 2: If f(x) − g¡0; z= o. 

At the optimal solution d will be forced into d = g − f(x), resulting in ZZ =0=Z. 

This proposition is then proved. 

Referring to Proposition 1, expression (4.2) can be linearized as follows: 

where 

(log a12)=5:67887(log a12 − log 2) − 20:61605(log a12 − log 3+d12) 

= −14:93718 log a12 − 20:61605d12 +8:126845435; 

log a12 − log 3+d12 ¿ 0; a12;d12 ¿ 0: (4.3) 

Hence, Example 1 can be formulated as the following program: 

Maximize (log a12) 

Subject to (log a12)=− 14:93718 log a12 − 20:61605d12 +8:126845435; 

log a12 − log 3+d12 ¿ 0; a12;d12 ¿ 0: 

By executing the above program on either LINDO [35] or EXCEL [36], the obtained solution 

set becomes ( (log a12)=1;d12 =0; log a12 =0:477121 and a12 = 3). 

From the basis of Propositions 1 and 2, the following corollary is presented: 

Corollary 1. A triangular fuzzy value log aij can be interpreted as 

log aij = { (log aij)+(sij;R − sij;L) log aij;2 − (sij;R − sij;L)dij + sij;L log aij;1}=sij;R; 

where log aij + log aij;2 + dij ¿ 0; aij;2;dij ¿ 0. 

Proof. Within Proposition 1, a triangle membership function (log aij) can be interpreted by 

(log aij)=sij;L(log aij − log aij;1)+(sij;R − sij;L)=2(|log aij − log aij;2| + log aij − log aij;2).


By applying Proposition 2, (log aij) can then be linearized as 

where 

(log aij)=sij;R log aij − (sij;R − sij;L) log aij;2 +(sij;R − sij;L)dij − sij;L log aij;1; 

log aij − log aij;2 + dij ¿ 0 and aij;2;dij ¿ 0: (4.4) 

Accordingly, after inverting (log aij) and log aij in (4.4), 

log aij = { (log aij)+(sij;R − sij;L) log aij;2 − (sij;R − sij;L)dij + sij;L log aij;1}=sij;R; 

where log aij + log aij;2 + dij ¿ 0; aij;2;dij ¿ 0. 

By doing so, this corollary is completed. 

View (4.3) as an instance. After reversing (log a12) and log a12, 

log a12 = − 0:06694704 (log a12) − 1:380183542d12 +0:54406825351; 

where log a12 − log 3+d12 ¿ 0; a12;d12 ¿ 0. 

Herein LLSM was employed to reduce the fractional form wi=wj in Problem 1 to linear form 

log wi − log wj. Axioms 1, 2 and Theorem 2 are thereby introduced to prove that all decision 

contents retain their original meaning after transformation of a fuzzy number to a log value. 

Axiom 1. For a fuzzy number x, let xL; xM; xU are the lower, modal, and upper values of 

x, respectively. In a restricted domain [xL;xU], the possibility of (xL) is 0, the possibility of 

(xM) is 1, and the possibility of (xU) is 0. The slope between (xL) and (xM) is increasing 

while the slope between (xM) and (xU) is decreasing where xL ¡xM ¡xU. 

Axiom 2. After transforming a fuzzy number x to a log value, let log xL; log xM; log xU represent 

the lower, modal, and upper values of log x, respectively. In a restricted domain [log xL; log xU], 

the possibilities of (log xL); (log xM), and (log xU) are separately 0; 1, and 0. The slope 

between (log xL) and (log xM) increases while the slope between (log xM) and (log xU) 

decreases where log xL ¡ log xM ¡ log xU. 

Theorem 2. An optimal solution for the problem PP1 is the same as that for PP2; PP1 and 

PP2 are presented below in which xL; xM; xU are the lower; modal; and upper values of fuzzy 

number x; respectively; 

PP1: Maximize x 

Subject to x ¿ 0; 

where x is a fuzzy number depicted in Fig:3a: 

PP2: Maximize log x 

Subject to x ¿ 0; 

where log x is a fuzzy number depicted in Fig:3b:


Fig. 3. (a) Afuzzy number x; (b) a fuzzy number log x. 

Proof. Based on Axioms 1 and 2, the value of each membership function ranges from zero to 

one and the unit change in x results in proportional change in log x guided by proportionally 

adjusted slopes. This implies that the increasing or decreasing slopes are di erent between (x) 

and (log x), but 

(i) if the optimal value x of (x) or (log x) is between xL and xM, both PP1 and PP2 desire 

to maximize x until x or log x reach the constrained limit of x value; 

(ii) if the optimal value x of (x) or (log x) is between xM and xU, both PP1 and PP2 desire 

to minimize x until x or log x reach the constrained limit of x value. 

Since PP1 and PP2 encounter the same constraints, the found optimal value x for PP1 is identical 

to that for PP2. This theorem is thus proved. 

Corollary 1 only interprets triangular membership functions. Subsequently, Proposition 3 is 

proposed to represent general separable linear membership functions. 

Proposition 3. Fig. 4 depicts that (log aij) is a general separable linear membership function 

of a fuzzy value log aij; where log aij;k; k=0; 1; 2;:::;m − 1; are the change points of 

(log aij);sij;k; k=1; 2;:::;m− 1; are the slopes of line segments between log aij;k−1 and log 

aij;k; and 

sij;k = (log aij;k) − (log aij;k−1) 

: 

log aij;k − log aij;k−1 

(log aij) can then be expressed as 

(log aij)=sij;1(log aij − log aij;0) 

m−1 � 

+ 

k=2 

sij;k − sij;k−1 

(|log aij − log aij;k−1| + log aij − log aij;k−1): 

2


Fig. 4. Ageneral separable linear membership function. 

Proof. This proposition can be inspected as follows: 

(i) If log aij 6 log aij;1 then 

(log aij)=sij;1(log aij − log aij;0)+ sij;2 − sij;1 


2 

= sij;1(log aij − log aij;0): 

(ii) If log aij;1 6 log aij 6 log aij;2 then 

(log aij)=sij;1(log aij − log aij;0)+ sij;2 − sij;1 


2 

+ sij;3 − sij;2 


2 

= sij;1(log aij − log aij;0)+ sij;2 − sij;1 

(|log aij − log aij;1| + log aij − log aij;1): 

2 

(iii) If log aij;k ′ −1 6 log aij 6 log aij;k ′ then �m−1 k¿k ′(|log aij −log aij;k ′ −1 | +log aij −log aij;k ′ −1)=0 

and (log aij) becomes sij;1(log aij −log aij;0)+ � k ′ −1 

k=2 (sij;k −sij;k−1)=2(|log aij −log aij;k−1|+ 

log aij − log aij;k−1). 

This proposition is then proved. Consider the following example. 

Example 2. 

Maximize (log a 1 24) 

Subject to a 1 24 ¿ 0; 

where log a 1 24 

is a fuzzy value depicted in Fig. 5.


Based on Proposition 3, (log a 1 24 

Fig. 5. Afuzzy value log a 1 24. 

) can be expressed below: 

(log a1 � 

24)=2:65754 log a1 � �� 

1 

24 − log 

3 

+ 1:13578 − 2:65754 

�� 

�� 

log a 

2 

1 � �� 

2 �� 

24 − log + log a 

3 

1 � �� 

2 

24 − log 

3 

+ −1:13578 − 1:13578 

+ −11:94979 + 1:13578 

(|log a 

2 

1 24 − log 1| + log a1 24 − log 1) 

�� 

�� 

log a1 24 − log 

� 

=2:65754 log a1 � 

1 

24 − log 

3 

+ −1:52177 

2 

2 

�� 

�� log a 1 24 − log 

� 2 

3 

� 3 

2 

�� + log a 1 24 − log 

+ −2:27155 

(|log a 

2 

1 24 − log 1| + log a1 24 − log 1) 

�� 

�� 

log a1 24 − log 

+ −10:81401 

2 

� 3 

2 

�� + log a 1 24 − log 

�� + log a 1 24 − log 

� �� 

2 

3 

� �� 

3 

2 

� �� 

3 

: (4.5) 

2


To linearize a group of absolute terms with negative coe cients, the following proposition 

is introduced. 

Proposition 4. Consider the following problem: 

m−1 � ck 

PP3: Maximize z = 

2 (|log aij − log aij;k−1| + log aij − log aij;k−1) 

k=2 

Subject to log aij ∈ F (a feasible set); aij ¿ 0; ck 6 0; 

can be linearized as PP4 below: 

m−1 � 

PP4: Maximize w = 

where ck is a negative coe cient; k=2; 3;:::;m−1 and log aij; 

m − 1 ¿ log aij;m−2 ¿ ···¿ log aij;2 ¿ log aij;1 ¿ log aij;0 

k=2 

ck 

� 

�k−1 

log aij − log aij;k−1 + 

m−2 � 

Subject to log aij + dij;‘ ¿ log aij;m−2; 

‘=1 

‘=1 

dij;‘ 

log aij ∈ F(a feasible set); aij ¿ 0; ck 6 0: 

Proof. According to Proposition 2, PP3 is equivalent to following program: 

m−1 � 

PP5: Maximize w = ck(log aij − log aij;k−1 + rij;k−1) 

k=2 

Subject to log aij−log aij;k−1+rij;k−1¿0; rij;k−1¿0 for k =2; 3;:::;m−1; 

log aij ∈ F(a feasible set); aij ¿ 0; ck 6 0; 

where rij;k−1 is a deviation variable: 

PP5 implies that: 

if log aij ¡ log aij;k−1 then at optimal solution rij;k−1 = log aij;k−1 − log aij; 

if log aij ¿ log aij;k−1 then at optimal solution rij;k−1 =0. 

�


Substitute rij;k−1 by � k−1 

m−1 � 

PP6: Maximize z = 

‘=1 dij;‘, PP5 then becomes 

k=2 

ck 

� 

�k−1 


Subject to log aij + dij;1 ¿ log aij;1; 

log aij + dij;1 + dij;2 ¿ log aij;2; 

. 

. 

. 

. 

‘=1 

dij;‘ 

log aij + dij;1 + dij;2 + ···+ dij;m−3 ¿ log aij;m−3; 

log aij + dij;1 + dij;2 + ···+ dij;m−2 ¿ log aij;m−2; 

log aij ∈ F(a feasible set); aij ¿ 0; ck 6 0: 

Since the rst (m − 2) constraints display log aij ¿ log aij;m−2 − � m−2 

‘=1 dij;‘ ¿ log aij;m−3 − 

� m−3 

‘=1 dij;‘ ¿ ···¿ log aij;2 − dij;1 − dij;2 ¿ log ij;1 − dij;1 ¿ 0, the rst (m − 3) constraints in PP6 

are covered by the (m − 2)th constraint in PP6. As a result, Proposition 4 is proved. 

Hence, consider (log a 1 24 

where 

) in Example 2, expression (4.5) can be linearized as 

(log a1 � 

24)=2:65754 log a1 � �� 

1 

24 − log − 1:52177 log a 

3 

1 � 

2 

24 − log 

3 

−2:27155(log a 1 24 − log 1+d 1 24;1 + d 1 24;2) 

−10:81401 

� 

log a 1 24 − log 

� 

3 

2 

� 

� 

+ d 1 24;1 + d 1 24;2 + d 1 24;3 

= −11:94979 log a 1 24 − 14:60733d 1 24;1 − 13:08556d 1 24;2 

−10:81401d 1 24;3 +2:90424; 

� 

� 

+ d 1 24;1 

log a 1 24 + d 1 24;1 + d 1 24;2 + d 1 24;3 ¿ 0:17609: (4.6) 

Nevertheless, Example 2 can be reformulated as 


Subject to (log a 1 24)=− 11:94979 log a 1 24 − 14:60733d 1 24;1 − 13:08556d 1 24;2 

−10:81401d1 24;3 +2:90424 

� � 

3 

=0:17609; a 

2 

1 24 ¿ 0: 

log a 1 24 + d 1 24;1 + d 1 24;2 + d 1 24;3 ¿ log 

�


By using LINDO [35] or EXCEL [36], the obtained solution set is ( (log a 1 24 )=1, log a1 24 =0, 

a 1 24 =1, d1 24;1 =0, d1 24;2 = 0, and d1 24;3 =0:17609). 

Similar to Corollary 1, Corollary 2 is described as follows. 

Corollary 2. A general concave fuzzy value log aij can be interpreted as 

� 

m−2 � 

m−1 � 

log aij = (log aij) − (sij;m−1 − sij;k)dij;k + 

k=1 

where log aij + � m−2 

‘=1 dij;‘ ¿ log aij;m−2; aij ¿ 0; and sij;0 =0. 

k=1 

(sij;k − sij;k−1)log aij;k−1 

�� 

(sij;m−1); 

Proof. Regarding Proposition 3, a piecewise linear membership function (log aij) can be represented 

by 

m−1 � 

sij;1(log aij − log aij;0)+ 

k=2 

sij;k − sij;k−1 


2 

After utilizing Proposition 4 to linearize the absolute terms with negative coe cients, (log aij) 

can then be interpreted as 

� 

� 

m−1 � 

�k−1 

(log aij)=sij;1(log aij − log aij;0)+ (sij;k − sij;k−1) log aij − log aij;k−1 + ; 


‘=1 dij;‘ ¿ log aij;m−2 and aij ¿ 0. 

Accordingly, after expanding (2:7), the following occurs: 

k=2 

(log aij)=sij;1 × log aij − sij;1 × log aij;0 +(sij;2 − sij;1)(log aij − log aij;1 + dij;1) 

+(sij;3 − sij;2)(log aij − log aij;2 + dij;1 + dij;2) 

‘=1 

dij;‘ 

(4.7) 

+(sij;4 − sij;3)(log aij − log aij;3 + dij;1 + dij;2 + dij;3)+···+(sij;m−1 − sij;m−2) 

� 

m−2 � 

log aij − log aij;m−2 + 

� 

: (4.8) 

‘=1 

dij;‘ 

As a result, after reorganizing (4.8), we obtain 

m−2 � 

m−1 � 

(log aij)= (sij;m−1 − sij;k)dij;k − (sij;k − sij;k−1)log aij;k−1 + sij;m−1 × log aij: (4.9) 

k=1 

k=1 

After reversing (log aij) and log aij in (4.9), 

log aij = 

� 

m−2 � 

(log aij) − 

k=1 

� 

m−1 

(sij;m−1 − sij;k)dij;k + 


‘=1 dij;‘ ¿ log aij;m−2;aij ¿ 0, and sij;0 =0. 

Therefore, Corollary 2 is completed. 

k=1 


�� 

(sij;m−1);



Take (4.6) as an example. After switching (log a24) and log a24, 

log a 1 24 =[ (log a 1 24)+14:60733d 1 24;1 +13:08556d 1 24;2 

+10:81401d1 24;3 − 2:90424)]=(−11:94979) 

where log a1 24 + d124;1 + d124;2 + d124;3 ¿ 0:17609. 

In reality, fuzzy ratings obtained from the decision-maker(s) may be concave, convex, or 

concave–convex mixed membership functions. Since Corollary 2 cannot treat convex membership 

functions, the proposition of treating a convex membership function is introduced below. 

Consider the following example. 

Example 3. 


Subject to a 2 24 ¿ 0; 


is a convex–concave mixed fuzzy value depicted in Fig. 6. 

) can be expressed as follows: 

(log a2 � 

24)=1:99316 log a2 � �� 

1 

24 − log + 

3 

0:27839 

�� 

�� 

log a 

2 

2 � �� 

2 �� 

24 − log 

3 

� �� 

2 

Referring to Proposition 3, the (log a 2 24 

+ log a 2 24 − log 

3 

+ −3:40732 

(|log a 

2 

2 24 − log 1|


+log a2 24 − log 1) −10:81401 

�� 

�� 

log a 

2 

2 � �� 

3 �� 

24 − log + log a 

2 

2 � �� 

3 

24 − log : 

2 

Note that (log a2 1 

24 ) is a convex curve between log( 3 ) and log 1 and (log a224 curve between log( 2 

7 

3 ) and log( 4 ). That is, (log a224 (4.10) 

) is a concave 

) is a convex–concave mixed membership 

function. Since Propositions 2 and 4 can only linearize the absolute terms with negative coefcients, 

the linearization of the absolute term with a positive coe cient is presented. 

Proposition 5. Consider the following program 

� 

Maximize z = c 

2 (|log aij − log aij;k−1| + log aij − log aij;k−1) 

� 

Subject to log aij ∈ F; aij ¿ 0;c¿ 0; and log aij;k−1 is a constant 

can be linearized as the program below: 

{Maximize zz = c(log aij − yij;k−1 + vij;k−1log aij;k−1 − log aij;k−1) 

Subject to yij;k−1 ¿ log aij +(vij;k−1 − 1)M; log aij ∈ F; aij;yij;k−1;c¿ 0; 

vij;k−1 is a zero–one variable; M is a big value:} 

Proof. 

Case 1: If log aij − log aij;k−1 ¿ 0;z= c(log aij − log aij;k−1). 

At the optimal solution vij;k−1 = 0 then yij;k−1 = 0, which results in zz = c(log aij − 

log aij;k−1)=z. 

Case 2: If log aij − log aij;k−1 ¡ 0;z=0. 

At the optimal solution vij;k−1 = 1 then yij;k−1 = log aij, which results in zz =0=z. 

This proposition is then proved. 

Accordingly, based on Propositions 4 and 5, the (log a2 24 ) in (4.10) can be linearized as 

follows: 

(log a2 � 

24)=1:99316 log a2 � �� 

1 

24 − log 

3 

+ 0:27839 

� 

2 log a 

2 

2 24 − 2y2 24;1 +2v2 � � � �� 

2 2 

24;1log − 2 log 

3 3 

−3:40732(log a2 24 − log 1+d2 24;2) − 10:81401(log a2 � � 

3 

24 − log +d 

2 

2 24;2+d2 24;3) 

= −11:94979 log a 2 24−0:27839y 2 24;1−0:04902v 2 24;1−14:22133d 2 24;2 

−10:81401d 2 24;3+2:90425; (4.11)


where y2 24;1 ¿ log a224 +(v2 24;1 − 1)M; log a224 + d224;2 + d224;3 ¿ 0:17609; a224 ;y2 24;1 ¿ 0, v2 24;1 is a 

zero–one variable, M is a big value. 

Hence, Example 3 can be reprogrammed as follows: 


Subject to (log a 2 24)=− 11:94979 log a 2 24 − 0:27839y 2 24;1 − 0:04902v 2 24;1 − 14:22133d 2 24;2 

−10:81401d2 24;3 +2:90425 

y2 24;1 ¿ log a2 24 +(v2 24;1 − 1)M; log a2 24 + d2 24;2 + d2 24;3 ¿ 0:17609; 

a 2 24;y 2 24;1 ¿ 0; v 2 24;1 is a 0–1 variable; M is a large value: 

After computing with EXCEL, the obtained solution set is ( (log a 2 24 )=1; log a2 24 =0;a2 24 =1, 

v 2 24;1 =0;y2 24;1 =0;d2 24;2 =0;d3 24;3 =0:17609). 

Similar to Corollaries 1 and 2, Corollary 3 is described as follows: 

Corollary 3. A general concave and convex mixed fuzzy value log aij can be interpreted as 

� 

� 

�k−1 

log aij = (log aij) − 

(sij;k − sij;k−1) 

where 

− 

m−2 

log aij + 

� 

for k where sij; k¿sij; k−1 

m−1 � 

+ 

k=1 

for k where sij; k¡sij; k−1 


� 

dij;k ¿ log aij;m−2 

k=1 

‘=1 

dij;‘ 

(sij;k − sij;k−1)(vij;k−1log aij;k−1 − yij;k−1) 

�� 

(sij;m−1); 

for sij;k ¡sij;k−1 (concave segments); 

0 6 dij;k 6 log aij;k − log aij;k−1 

and 

yij;k−1 ¿ log aij +(vij;k−1 − 1)M 

for sij;k ¿sij;k−1 (convex segments): 

yij;k−1 ¿ 0 

Referring to Proposition 3, a general separable linear membership function (log aij) can be 

interpreted by 

m−1 � sij;k − sij;k−1 

sij;1(log aij − log aij;0)+ 


2 

k=2 

After using Propositions 4 and 5, (log aij) can then be interpreted as 

� 

(log aij)=sij;1(log aij − log aij;0)+ 


for k where sij; k¡sij; k−1

where 

m−2 

log aij + 


� 

�k−1 


‘=1 

dij;‘ 

� 

+ 

� 

for k where sij; k¿sij; k−1 


(log aij − log aij;k−1 + vij;k−1log aij;k−1 − yij;k−1); (4.12) 

� 

dij;k ¿ log aij;m−2 

k=1 

0 6 dij;k 6 log aij;k − log aij;k−1 

yij;k−1 ¿ log aij +(vij;k−1 − 1)M 

yij;k−1 ¿ 0 

for sij;k ¡sij;k−1 (concave segments); 

for sij;k ¿sij;k−1 (convex segments); 

and aij;yij;k−1 ¿ 0; vij;k−1 are 0–1 variables, and M is a large value. 

Accordingly, after expanding (4.12) and inverting (log aij) and log aij in (4.12), this corollary 

is proved. 

Take (4.11) as an instance. By reversing (log a 2 24 ) and log a2 24 

in (4.11), 

log a 2 24 = { (log a 2 24)+0:27839y 2 24;1 +0:04902v 2 24;1 +14:22133d 2 24;2 +10:81401d 2 24;3 

−2:90425}=(−11:94979); 

where y 2 24;1 ¿ log a2 24 +(v2 24;1 − 1)M; log a2 24 + d2 24;2 + d2 24;3 ¿ 0:17609; a2 24 ;y2 24;1 ¿ 0; v2 24;1 isa0 

–1 variable, and M is a large value. 

5. Proposed models 

By incorporating a fuzzy number expression and an absolute term linearization technique into 

a GP-AHP method, the proposed method involves the trade-o consideration of optimizing the 

decision-makers’ group inconsistency as well as individual desires. The derived corresponding 

priority vector best re ects the majority of the involved individual’s preference and is progressively 

less sensitive and vulnerable to con icting group judgments. Consequently, by utilizing 

Corollary 1 to treat triangular fuzzy values and Propositions 2 and 4 to linearize absolute terms, 

Problem 4 can be reprogrammed as the following model.


5.1. The proposed GP-AHP model (1) 

Minimize 

⎧⎡ 

⎨ n� 

⎣ 

⎩ 

n� 

i=1 j¿i e=1 

Subject to log vi − log vj − log a e ij + e ij ¿ 0; 

E� 

(log vi − log vj − log ae ij +2 e ⎤ 

ij) ⎦ − 

n� 

n� 

E� 

i=1 j¿i e=1 

log a e ij = { (log a e ij)+(s e ij;R − s e ij;L) log a e ij;2 − (se ij;R − s e ij;L)d e ij 

+ s e ij;L log a e ij;1 }=se ij;R 

(log ae ⎫ 

⎬ 

ij) 

⎭ 

log a e ij + log a e ij;2 + d e ij ¿ 0 vi;a e ij; e ij;d e ij ¿ 0(i; j) ∈{(i; j) | 1 6 i¡j6 n}; 

where =(n × E)=|M 1 − M 0 |, the un-normalized vector V =(v1;:::;vn) will be normalized to 

produce the vector W =(w1;w2;:::;wn) with vi=vj = wi=wj, and � 

i wi =1. 

Notably the proposed GP-AHP model (1) considers only the two-level structured AHP problems. 

In reality an AHP problem may involve comparing criteria and alternatives in a three-level 

structure or higher. Therefore, the following model is presented. 


Minimize 

⎧ 

⎨ 

⎩ 

m� 

m� 

q=1 q ′ ¿q e=1 

+ 

m� 

⎧⎡ 

⎨ m� 

− ⎣ 

⎩ 

n� 

E� 

[(log vq − log vq ′) − log aeqq ′ +2 e qq ′] 

n� 

q=1 i=1 j¿i e=1 

m� 

E� 

E� 

[(log vqi − log vqj) − log ae qij +2 e ⎫ 

⎬ 

qij] 

⎭ 

(log a e qq ′)+ 

q=1 q ′ ¿q e=1 

q=1 i=1 j¿i e=1 

Subject to log vq − log vq ′ − log aeqq ′ + e qq ′ ¿ 0; log vqi − log vqj − log ae qij + e qij ¿ 0; 

m� 

n� 

n� 

E� 

(log ae ⎤⎫ 

⎬ 

qij) ⎦ 

⎭ 

log a e qq ′ = { (log ae qq ′)+(se qq ′ ;R − se qq ′ ;L ) log ae qq ′ ;2 − (se qq ′ ;R − se qq ′ ;L )de qq ′ 

+ s e qq ′ ;L log ae qq ′ ;1 }=se qq ′ ;R 

log ae qq ′ + log aeqq ′ ;2 + deqq ′ ¿ 0; 

log a e qij = { (log a e qij)+(s e qij;R − s e qij;L) log a e qij;2 − (se qij;R − s e qij;L)d e qij 

+ s e qij;L log a e qij;1 }=se qij;R 

log a e qij + log a e qij;2 + d e qij ¿ 0; vq;vqi;a e qq ′;ae qij; e qq ′; e qij;d e qq ′;de qij ¿ 0; 

(q; q ′ ) ∈{(q; q ′ ) | 1 6 q¡q ′ 6 m}; (i; j) ∈{(i; j) | 1 6 i¡j6 n};


where ae qq ′ represents the eth decision-maker’s uncertainty assessment to which criterion q is 

preferred more than criterion q ′ , ae qij represents the eth decision-maker’s uncertainty assessment 

to which object i is preferred over object j under the qth criterion, m is the number of criteria, 

n is the number of alternatives, E is the number of decision makers, and =(m + m × n) × 

E=|M 1 − M 0 |. 

Note that the proposed models (1) and (2) can only solve fuzzy AHP problems with triangular 

fuzzy ratings. In practice an AHP problem may have general concave fuzzy estimates rather 

than just triangular fuzzy estimates. Hence, by using Corollary 2 to treat more general concave 

fuzzy judgments, the following model is formulated. 


Minimize 

⎧ 

⎨ 

⎩ 

m� 

m� 

q=1 q ′ ¿q e=1 

+ 

m� 

⎧⎡ 

⎨ m� 

− ⎣ 

⎩ 

n� 

E� 


n� 

q=1 i=1 j¿i e=1 

m� 

E� 

E� 


⎬ 

qij] 

⎭ 


q=1 q ′ ¿q e=1 

q=1 i=1 j¿i e=1 


log ae qq ′ = 

log a e qij = 

� 

� 

(log ae m−2 � 

qq ′) − 

loga e qq ′ ;k−1 

k=1 

�� 

m� 

n� 

n� 

E� 

� 

(se qq ′ ;m−1 − se qq ′ ;k )deqq ′ ;k + 

m−1 

se qq ′ ;m−1 log aeqq ′ + 

� 

k=1 


⎬ 

qij) ⎦ 

⎭ 

(s e qq ′ ;k − se qq ′ ;k−1 ) 

m−2 

de qq ′ ;‘ ¿ log aeqq ′ ;m−2 ; 

‘=1 

(log ae m−2 � 

qij) − (se qij;m−1 − se qij;k )deqij;k + 

m−1 � 

(se qij;k − se qij;k−1 ) 

loga e qij;k−1 

k=1 

k=1 

�� 

se qij;m−1 log ae m−2 � 

qij + de qij;‘ ¿ log aeqij;m−2 ; 

‘=1 

vq;vqi;a e qq ′;ae qij; e qq ′; e qij;d e qq ′;de qij ¿ 0; 

(q; q ′ ) ∈{(q; q ′ ) | 1 6 q¡q ′ 6 m}; (i; j) ∈{(i; j) | 1 6 i¡j6 n}; 

where s e qq ′ ;k stands for the slope between line segments log ae qq ′ ;k and log ae qq ′ ;k+1 , se qij;k stands 

for the slope between lines segments log a e qij;k and log ae qij;k+1 .


In real world AHP problems, a fuzzy rating should not be limited to concave shape or 

convex shape only. Using Corollary 3 to treat concave, convex, or concave–convex mixed 

fuzzy evaluations, this work introduces the following model. 


Minimize 

⎧ 

⎨ 

⎩ 

m� 

m� 

q=1 q ′ ¿q e=1 

+ 

m� 

⎧⎡ 

⎨ m� 

− ⎣ 

⎩ 

n� 

E� 


n� 

q=1 i=1 j¿i e=1 

m� 

E� 

E� 


⎬ 

qij] 

⎭ 


q=1 q ′ ¿q e=1 

q=1 i=1 j¿i e=1 


log ae ⎧ 

⎨ 

qq ′ = 

⎩ (log aeqq ′) − 

− 

log ae m−2 � 

qq ′ + 

k=1 

� 

m� 

for k where s e 

qq ′ ;k ¿se 

qq ′ ;k−1 

−ye qq ′ ;k−1 )+ 

m−1 � 

n� 

� 

n� 

E� 

for k where s e 

qq ′ ;k ¡se qq;;k−1 

k=1 

d e qq ′ ;k ¿ log ae qq ′ ;m−2 

0 6 d e qq ′ ;k 6 log ae qq ′ ;k − log ae qq ′ ;k−1 

ye qq ′ ;k−1 ¿ log aeqq ′ +(ve qq ′ ;k−1 

ye qq ′ ;k−1 ¿ 0 

log ae ⎧ 

⎨ 

qij = 

⎩ (log aeqij) − 

− 

� 


⎬ 

qij) ⎦ 

⎭ 

� 

(se qq ′ ;k − se qq ′ ;k−1 ) 

k−1 

‘=1 

d e qq ′ ;‘ 

(s e qq ′ ;k − se qq ′ ;k−1 )(ve qq ′ ;k−1 log ae qq ′ ;k−1 

(s e qq ′ ;k − se qq ′ ;k−1 ) log ae qq ′ ;k−1 

− 1)M 

for k where s e qij; k ¿se qij; k−1 

� 

for k where s e qij; k ¡se qij; k−1 

for s e qq ′ ;k ¡se qq ′ ;k−1 ; 

for s e qq ′ ;k ¿se qq ′ ;k−1 ; 

(s e qij;k − se qij;k−1 ) 

⎫� 

⎬ 

(s 

⎭ 

e qq ′ ;m−1 ); 

(se qij;k − se qij;k−1 ) 

�k−1 

‘=1 

d e qij;‘

where v e qq ′ ;k and ve qij;k 


(ve qij;k−1 log aeqij;k−1 − ye qij;k−1 ) 

m−1 � 

+ (se qij;k − se qij;k−1 ) log ae ⎫� 

⎬ 

qij;k−1 ⎭ 

k=1 

log ae m−2 � 

qij + de qij;k ¿ log aeqij;m−2 k=1 

0 6 d e qij;k 6 log ae qij;k − log ae qij;k−1 

(s e qij;m−1); 

for s e qij;k ¡se qij;k−1 ; 

ye qij;k−1 ¿ log aeqij +(ve qij;k−1 − 1)M 

ye for s 

qij;k−1 ¿ 0 

e qij;k ¿se qij;k−1 ; 

vq;vqi;ae qq ′;ae e e 

qij; qq ′; qij;de qq ′;deqij;k ;ye qq ′ ;k ;ye qij;k ¿ 0; 

(q; q ′ ) ∈{(q; q ′ ) | 1 6 q¡q ′ 6 m}; (i; j) ∈{(i; j) | 1 6 i¡j6 n}; 

are 0–1 variables and M is a big value. 

6. Solution algorithm and numerical examples 

Based on the previous discussion, a solution algorithm is described as follows. 

6.1. Solution algorithm 

Step 1: Express each fuzzy comparison by using Corollaries 1, 2 or 3. 

Step 2: Formulate the problem by applying the proposed GP-AHP method. 

Step 3: Compute M 0 , M 1 , and values with Theorem 1. 

Step 4: Derive the vector V by employing any popular linear programming package like LINDO 

or EXCEL to solve the programmed model. 

Step 5: Generate the priority vector W by normalizing the vector V . 

Now consider the following three-level structured AHP problem initially provided by Laarhoven 

et al. [8]. 

Example 4. Assume that a professorship position is vacant in the Operations Research Department 

of a certain university. After several competitive screening interviews, only three serious 

candidates remain, referred to herein as A, B, and C. To identify which applicant is best quali 

ed for the job, the committee has been installed to provide advice. The committee has three 

members and they assess the candidates by four decision criteria: (1) mathematical creativity 

(q1); (2) creativity in implementations (q2); (3) administrative capabilities (q3); (4) maturity 

or personal integrity (q4). 

Therefore, the committee derives evaluations of the candidates following the above criteria. 

Through a pair-by-pair comparison, the relative importance of the decision criteria is constructed


Table 1 

The pairwise comparison of importance criteria evaluated by the committee 

q1 q2 q3 q4 

q1 1 ã 1 12 =( 2 

3 

ã 2 12 =( 2 

5 

ã 3 12 =( 3 

2 

q2 1=ã s 12 for all s 1 ã e 23 =( 5 

2 

3 ; 1; 2 ), ãe13 =( 2 3 

3 ; 1; 2 ) ã114 =( 2 1 

7 ; 3 

1 2 ; 2 ; 3 ), forall e ã214 =( 2 1 

7 ; 3 

5 ; 2; 2 ), ã314 =( 2 1 

5 ; 2 

; 3; 7 

2 ) ã124 =( 2 

3 

for all e ã 2 24 =( 2 

3 

ã 3 24 =( 3 

2 

q3 1=ã s 13 for all s 1=ã e 23 for all e 1 ã 1 34 =( 2 

3 

q4 1=ã s 14 for all s 1=ã e 24 for all e 1=ã e 34 for all e 1 

Table 2 

The pairwise comparison of candidates under criterion 1 

ã 2 34 =( 2 

3 

ã 3 34 =( 3 

2 

q =1 AB C 

; 2 

5 ), 

; 2 

5 ), 

; 2 

3 ) 

; 1; 3 

2 ), 

; 1; 3 

2 ), 

; 2; 5 

2 ) 

; 1; 3 

2 ), 

; 1; 3 

2 ), 

; 2; 5 

2 ) 

A 1 ã e q12 ã e q13 

B 1=ã e q12 1 ã e q23 

C 1=ã e q13 1=ã e q23 1 

Table 3 


q =2 AB C 

A 1 ã e q12 ã e q13 

B 1=ã e q12 1 — 

C 1=ã e q13 — 1 

in Table 1 containing fuzzy estimates. Following the AHP procedure, three candidates A, B, 

and C are compared under each of the criteria separately and Tables 2–5 summarize the 

assessed results. 

This example initially given by Laarhoven et al. [8] assumed that all membership functions 

of fuzzy ratings are triangular forms. Where e =1; 2; 3 represent each member’s evaluation, 

q =1; 2; 3; 4 represent each criterion, “—” means no information available, ã 1 1;12 =ã 2 1;12 =ã 3 1;12 = 

( 2 

3 

; 1; 3 

2 ), ã11;13 =ã 2 1;13 =( 2 3 

3 ; 1; 2 ), ã31;13 =( 2 1 2 

5 ; 2 ; 3 ), ã11;23 =ã 2 1;23 =ã 3 1;23 =( 2 1 2 

5 ; 2 ; 3 ), ã12;12 =ã 2 2;12 = 

7 ; 3; 2 ), ã12;13 =ã 2 2;13 =ã 3 2;13 =( 5 7 

2 ; 3; 2 ), ã13;12 =ã 2 3;12 =( 5 7 

2 ; 3; 2 ), ã33;12 =( 3 5 

2 ; 2; 2 ), ã13;13 = 

ã 3 2;12 =( 5 

2


Table 4 


q =3 AB C 

A 1 ã e q12 ã e q13 

B 1=ã e q12 1 ã e q23 

C 1=ã e q13 1=ã e q23 1 

Table 5 


q =4 AB C 

A 1 — ã e q13 

B — 1 ã e q23 

C 1=ã e q13 1=ã e q23 1 

ã 2 3;13 =ã 3 3;13 =( 5 

2 

; 3; 7 

ã 1 4;23 =ã 2 4;23 =ã 3 4;23 =( 3 

2 

2 ), ã13;23 =ã 2 3;23 =ã 3 3;23 =( 2 

3 

; 2; 5 

2 ). 


; 1; 3 

2 ), ã14;13 =ã 2 4;13 =( 3 

2 

; 2; 5 

2 ), ã34;13 =( 2 

5 

; 1 

2 

2 ; 3 ), and 

Based on the solution algorithm, the required six steps are: 

Step 1: For a triangular fuzzy value log a1 12 as displayed in Fig. 7, by employing Corollary 1, 

is expressed as follows: 

log a 1 12 

log a 1 12 = { (log a 1 12) − 11:35782 log 1+11:35782 d 1 12 +5:67891 log(2=3)}=(−5:67891) 

= −0:17609013 (log a 1 12) − 2 d 1 12 +0:176091259; (6.1) 

where log a1 12 − log 1+d112 ¿ 0, a112 ;d112 ¿ 0.


Similarly, each triangular fuzzy assessment can be formulated as given below: 

log a 2 12 = − 0:124940028 (log a 2 12) − 2:289237416 d 2 12 − 0:17608998; (6.2) 


1 − log( 2 )+d212 ¿ 0, a212 ;d212 ¿ 0; 

log a3 12 = − 0:096910023 (log a3 12) − 1:775652326 d3 12 +0:397939017; (6.3) 

where log a3 12 − log 2+d3 12 ¿ 0, a3 12 ;d3 12 ¿ 0; 

log ae 13 = − 0:17609013 (log ae 13) − 2de 13 +0:176091259; (6.4) 

where log ae 13 − log 1+de 13 ¿ 0, ae 13 ;de 13 ¿ 0, for e =1; 2; 3; 

log ae 14 = − 0:07918001 (log ae 14) − 2:18267383 de 14 − 0:397945039; (6.5) 

where log a e 14 

1 − log( 3 )+de 14 ¿ 0, ae 14 ;de 14 ¿ 0, for e =1; 2; 

log a3 14 = − 0:124940028 (log a3 14) − 2:289237416 d3 14 − 0:17608998; (6.6) 


1 − log( 2 )+d3 14 ¿ 0, a3 14 ;d3 14 ¿ 0; 

log ae 23 = − 0:066949999 (log ae 23) − 1:845541666 de 23 +0:544072297; (6.7) 


log ae 24 = − 0:17609013 (log ae 24) − 2 de 24 +0:176091259; (6.8) 

where log ae 12 − log 1+de 24 ¿ 0, ae 24 ;de 24 ¿ 0, for e =1; 2; 

log a3 24 = − 0:096910023 (log a3 24) − 1:775652326 d3 24 +0:397939017; (6.9) 


log ae 34 = − 0:17609013 (log ae 34) − 2 de 34 +0:176091259; (6.10) 


log a3 34 = − 0:096910023 (log a3 34) − 1:775652326 d3 34 +0:397939017; (6.11) 


log ae 112 = − 0:17609013 (log ae 112) − 2 de 112 +0:176091259; (6.12) 


log ae 113 = − 0:17609013 (log ae 113) − 2 de 113 +0:176091259; (6.13) 


log a3 113 = − 0:124940028 (log a3 113) − 2:289237416 d3 113 − 0:17608998; (6.14) 


1 − log( 2 )+d3 113 ¿ 0, a3 113 ;d3 113 ¿ 0; 

log a3 123 = − 0:124940028 (log a3 123) − 2:289237416 d3 123 − 0:17608998; (6.15) 


1 − log( 2 )+d3 123 ¿ 0, a3 123 ;d3 123 ¿ 0, for e =1; 2; 3; 

log a e 21j = − 0:066949999 (log a e 21j) − 1:8455416666 d e 21j +0:544072297; (6.16)


where log ae 21j − log 3+de 21j ¿ 0, ae 21j ;de 21j ¿ 0, for j =2; 3, and e =1; 2; 3; 

log ae 312 = − 0:066949999 (log ae 312) − 1:8455416666 de 312 +0:544072297; (6.17) 


log a3 312 = − 0:096910023 (log a3 312) − 1:775652326 d3 312 +0:397939017; (6.18) 


log ae 313 = − 0:066949999 (log ae 313) − 1:845541666 de 313 +0:544072297; (6.19) 


log ae 323 = − 0:17609013 (log ae 323) − 2 de 323 +0:176091259; (6.20) 


log ae 413 = − 0:096910023 (log ae 413) − 1:775652326 de 413 +0:397939017; (6.21) 


log a3 413 = − 0:124940028 (log a3 413) − 2:289237416 d3 413 − 0:17608998; (6.22) 


1 − log( 2 )+d3 413 ¿ 0, a3 413 ;d3 413 ¿ 0; 

log ae 423 = − 0:124940028 (log ae 423) − 2:289237416 de 423 − 0:17608998; (6.23) 

where log ae 423 − log 2+de 423 ¿ 0, ae 423 ;de 423 ¿ 0, for e =1; 2; 3. 

Step 2: In the form of the proposed model 2, Example 4 can be formulated as follows: 

⎧ 

⎨ 4� 4� 3� 

Minimize 

(log vq − log vq 

⎩ 

′ − log aeqq ′ +2 e qq ′) 

q=1 q ′ ¿q e=1 

+ 

⎧ 

⎨ 

− 

⎩ 

4� 

3� 

3� 

q=1 i=1 j¿i e=1 

4� 

4� 

3� 

q=1 q ′ ¿q e=1 

Subject to (6:1)–(6:23); 

3� 

(log vqi − log vqj − log ae qij +2 e ⎫ 

⎬ 

qij) 

⎭ 


4� 

3� 

3� 

3� 

q=1 i=1 j¿i e=1 

(log ae ⎫ 

⎬ 

qij) 

⎭ 

log vq − log vq ′ − log ae qq ′ + e qq ′ ¿ 0; log vqi − log vqj − log a e qij + e qij ¿ 0; 

vq;vqi;a e qq ′;ae qij; e qq ′; e qij;d e ij;d e qij ¿ 0; 

(q; q ′ ) ∈{(q; q ′ ) | 1 6 q¡q ′ 6 4}; (i; j) ∈{(i; j) | 1 6 i¡j6 3}: 

Step 3: Using Theorem 1 to obtain M 1 =2:25871, M 0 =0:9860733, and =47=(M 1 − M 0 )= 

37:71697: 

Step 4: After running on the LINDO or EXCEL, the acquired solution set is (log v1 =0:26061, 

log v2 =0:405622, log v3 =0, log v4 =0:21752, log v11 =0:175915, log v12 =0, log v13 =0:276157,


log v21 =0:476645, log v22 =0, log v23 =0, log v31 =0:476645, log v32 =0:028368, log v33 =0, 

log v41 =0:100242, log v42 =0:200485, log v43 =0; v1 =1:822258, v2 =2:544615, v3 =1, v4 = 

1:650137, v11 =1:499391, v12 =1,v13 =1:888674, v21 =2:996712, v22 =1,v23 =1,v31 =2:996712, 

v32 =1:0675, v33 =1, v41 =1:259627, v42 =1:586664 and v43 = 1) and the average grade of total 

membership functions is computed as 0.72461. 

Step 5: After calculating normalized the vector V , 

⎡ ⎤ 

v11 v12 v13 

⎢ ⎥ 

⎢ v21 v22 v23 ⎥ 

[v1;v2;v3;v4] ⎢ ⎥ 

⎢ ⎥ 

⎣ v31 v32 v33 ⎦ 

v41 v42 v43 

⎡ 

⎤ 

0:34170 0:22789 0:43041 

⎢ 

⎥ 

⎢ 0:59974 0:20013 0:20013 ⎥ 

=[0:25969; 0:36264; 0:14251; 0:23516] ⎢ 

⎥ 

⎢ 

⎥ 

⎣ 0:59174 0:2108 0:19746 ⎦ 

0:32749 0:41251 0:26000 

=[0:467567; 0:258803; 0:27363] = [w1;w2;w3] 

which are the weighting scores for the candidates A, B, and C, respectively. 

Example 5. To demonstrate the advantages of the proposed method, let Example 5 simultaneously 

encounter the pair-to-pair comparison involving triangular, general concave and concave– 

convex mixed fuzzy estimates, interval estimates, as well as some estimates with unavailable 

information. The fuzzy estimate ã 3 24 in Example 4 has been modi ed to an interval estimate that 

ranges between 3 5 

2 and 2 . The membership functions of fuzzy values ã124 and ã 2 24 in Example 4 

have been modi ed to be general concave and non-concave shaped as depicted in Figs. 5 and 

6, respectively. 

Based on the solution algorithm, the required six steps are: 

Step 1: Using Corollaries 2 and 3 to express the general concave and non-concave fuzzy 

ratings, respectively, 

log a 1 24 =[ (log a 1 24)+14:60733d 1 24;1 +13:08556d 1 24;2 +10:81401d 1 24;3 

−2:90424)]=(−11:94979) 

= −0:083683478 (log a 1 24) − 1:222392193d 1 24;1 − 1:095045185d 1 24;2 

− 0:904953978d 1 24;3 +0:243036906; (6.24) 

where log a1 24 + d124;1 + d124;2 + d124;3 ¿ 0:17609: 

log a 2 24 = { (log a 2 24)+0:27839y 2 24;1 +0:04902v 2 24;1 +14:22133d 2 24;2 +10:81401d 2 24;3 

− 2:90425}=(−11:94979)


= −0:083683478 (log a 2 24) − 0:023296643y 2 24;1 − 0:004102164v 2 24;1 − 1:19009d 2 24;2 

− 0:904953978d 2 24;3 +0:243037743; (6.25) 

where y2 24;1 ¿ log a224 +(v2 24;1 − 1)M, log a224 + d224;2 + d224;3 6 0:17609, 0 6 d224;2 6 0:17609, 

0 6 d2 24;3 ¿ 0:17609, a224 ;y2 24;1 ¿ 0, v2 24;1 is a 0–1 variable, M is a big value. 

Step 2: In the form of the proposed model 4, Example 5 is formulated as follows: 

Minimize 

+ 

⎧ 

⎨ 

⎩ 

⎧ 

⎨ 

− 

⎩ 

4� 

4� 

q=1 q ′ ¿q e=1 

4� 

3� 

3� 

q=1 i=1 j¿i e=1 

4� 

4� 

q=1 q ′ ¿q e=1 

3� 

(log vq − log vq ′ − log aeqq ′ +2 e qq ′) 

3� 

(log vqi − log vqj − log ae qij +2 e ⎫ 

⎬ 

qij) 

⎭ 

3� 


Subject to (6:1)–(6:7); (6:10)–(6:25); 

4� 

3� 

3� 

3� 

q=1 i=1 j¿i e=1 

(log ae ⎫ 

⎬ 

qij) 

⎭ 

log vq − log vq ′ − log ae qq ′ + e qq ′ ¿ 0; log vqi − log vqj − log a e qij + e qij ¿ 0; 

( 3 

2 ) 6 log a3 5 

24 6 ( 2 );vq;vqi;ae qq ′;ae e e 

qij; qq ′; qij;de ij;de qij ¿ 0; 

(q; q ′ ) ∈{(q; q ′ ) | 1 6 q¡q ′ 6 4}; (i; j) ∈{(i; j) | 1 6 i¡j6 3}: 

Step 3: Based on Theorem 1, the obtained M 1 is 2.05645, M 0 is 0.5751544, and value is 

31.72898104. 

Step 4: By executing the formulated model on LINDO or EXCEL, the acquired solution set 

is (log v1 =0:23571, log v2 =0:41353; log v3 =0, log v4 =0:21752, log v11 =0:17592, log v12 =0, 

log v13 =0:27616, log v21 =0:47664, log v22 =0, log v23 =0, log v31 =0:47664, log v32 =0, 

log v33 =0, log v41 =0:10024, log v42 =0:20049, log v43 =0, v1 =1:72072, v2 =2:59162, v3 =1, 

v4 =1:65014, v11 =1:49939, v12 =1, v13 =1:88869, v21 =2:99671, v22 =1, v23 =1, v31 =2:99671, 

v32 =1, v33 =1, v41 =1:25963, v42 =1:58667 and v43 = 1), where the average grade of total 

membership functions is computed as 0.74416. 

Step 5: After calculating normalized the vector V , 

⎡ 

⎢ 

[v1;v2;v3;v4] ⎢ 

⎣ 

v11 v12 v13 

v21 v22 v23 

v31 v32 v33 

v41 v42 v43 

⎤ 

⎥ 

⎦


Table 6 

The comparative results of solving Examples 4 and 5 

Example 4 Example 5 

Laarhoven et al. 

method [8] 

Buckley method 

[3] 

Boender et al. 

method [2] 

Chang method 

[4] 

Ruoning et al. 

[21] 

The proposed 

method 

The nal scores are fuzzy values (0:227; 0:398; 0:705), 

(0:168; 0:313; 0:579), (0:188; 0:289; 0:504) for Candidates 

A, B, and C, respectively 


(0:174; 0:332; 0:498), (0:187; 0:293; 0:476) for Candidates 

A, B, and C, respectively 


(0:22; 0:31; 0:42), (0:23; 0:29; 0:52) for Candidates A, B, 

and C, respectively 

The nal scores are crisp values 0:41; 0:28; 0:25 for 

Candidates A, B, and C, respectively 


(0:20; 0:29; 0:43), (0:21; 0:32; 0:51) for Candidates A, B, 

and C, respectively 

In the above methods the average grades of the total 

membership functions are unavailable 

The derived scores are crisp values 

0:467567; 0:258803; 0:2736 for Candidates A, B, 

and C, respectively. The average grade of total 

membership functions is 0.72461 

⎡ 

⎤ 

0:34170 0:22789 0:43041 

⎢ 

⎥ 

⎢ 0:59974 0:20013 0:20013 ⎥ 

=[0:24714; 0:37223; 0:14363; 0:23700] ⎢ 

⎥ 

⎢ 

⎥ 

⎣ 0:59974 0:20013 0:20113 ⎦ 

0:23749 0:41252 0:25999 

=[0:45722; 0:26444; 0:27834] = [w1;w2;w3] 

Their methods cannot treat Example 5 

The derived scores are crisp values 

0:45722; 0:26444; 0:27834 for Candidates 

A, B, and C, respectively. The 

average grade of total membership 

functions is 0.74416 

which are the weighting scores for the candidates A; B; and C; respectively: 

The comparison analysis in solving Examples 4 and 5 between traditional fuzzy AHP methods 

and the proposed method are summarized in Table 6. 

As was already shown by many researchers [2–4,8,21,37–44], one suitable method for dealing 

with fuzzy AHP problems involving fuzzy, point, and interval ratings as well as some 

ratings without available information is the fuzzy LLSM. The comparative analysis of Table 6


therefore focuses on this method while the other traditional fuzzy AHP methods cannot tackle 

this type of problems. 

Compared with conventional fuzzy AHP methods, numerical examples illustrate that the proposed 

GP-AHP method can concurrently treat the pair-to-pair comparison involving triangular, 

general concave and concave–convex mixed fuzzy estimates. By encompassing the trade-o 

consideration of optimizing decision-makers’ group inconsistency as well as individual opinion, 

the proposed method directly treats the interactions among the fuzzy variables and the 

decision-makers. The extracted corresponding priority vector therefore best re ects an overall 

preference and is progressively less sensitive and vulnerable to con icting judgments. 

7. Conclusions 

Due to the di culty that a decision-maker faces in precisely assessing the relative importance 

of two objectives, this study extends AHP practical applications to tackle a broader range of 

fuzzy problems by developing a separable linear expression to represent general triangular, concave, 

convex, and concave–convex mixed vague ratings. Since the developed solution algorithm 

is simple and the problem formulation means is distinct, a user-friendly computer program can 

be easily developed to handle simple yet time-consuming linearization calculations. Moreover, 

many prevailing software packages like LINDO [35] or EXCEL [36] can conveniently compute 

the formulated models. As a result, the generated priority vector of the proposed method allows 

the decision-maker to perform sensitivity analysis. Hence, the proposed method is a promising 

and attractive alternative to the current fuzzy AHP methods. 

Acknowledgements 

This research is supported by the National Science Council of the Republic of China under 

contract NSC 89-2416-H-158-008. 

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Chian-Son Yu is an Associate Professor in the Department of Information Management at Shih Chien University, 

Taipei. His research interests and publications are in the areas of fuzzy programming, fractional programming, 

robust programming, and global supply chain management. He received his Ph.D. in Information Management from 

National Chiao Tung University, Taiwan.

A GP-AHP method for solving group decision-making fuzzy AHP ...

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