IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

427 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

GROUP INTERVAL WEIGHTS BASED ON CONJUNCTION 

APPROXIMATION OF INDIVIDUAL INTERVAL WEIGHTS 

  
Tomoe Entani   

Graduate School of Applied Informatics   

University of Hyogo   

Kobe, JAPAN   

E-mail:entani@ai.u-hyogo.ac.jp 

 

ABSTRACT 

 

 

The individual and group decisions in this study are denoted as the normalized interval 

weights of alternatives as in Interval AHP. It assumes that a decision maker uses crisp 

values in the interval weights in giving comparisons. The interval weights reflect 

uncertainty in a decision maker’s mind. Then, the group interval weight is obtained as a 

conjunction approximation of the individual interval weights. For a consensus, the group 

interval weight is obtained so as to intersect with all the individual interval weights. In 

other words, the group interval weight has something in common with each individual 

interval weight. The group decision depends on how much the decision makers are 

satisfied or dissatisfied with it. The satisfaction of a decision maker is measured by the 

ranges of the group interval weights which s/he can support. Similarly, the decision 

maker’s dissatisfaction is defined by the ranges which are out of his/her decision. It is 

better to maximize the satisfaction and simultaneously to minimize the dissatisfaction. 

However, there is a trade-off between these two objectives. In the proposed model, the 

importance of the satisfaction or dissatisfaction is given. Then, the decision makers find 

not only the group decision but also their satisfaction and dissatisfaction with it.  

 

Keywords: Group decision making; interval analysis; Analytic Hierarchy Process 

 

 

1. Introduction 

In the Analytic Hierarchy Process (AHP), the crisp priority weights of alternatives are 

obtained from the pairwise comparison matrix given by a decision maker (Saaty, 1980). 

The elements of a matrix are crisp values such as 1/3, and 5. The well-known techniques 

to obtain the weights from the given crisp comparisons are geometric mean and 

eigenvector methods. The obtained weights are also crisp. The other technique is Interval 

AHP, where the weights are obtained as an interval to reflect the inconsistency among the 

given crisp comparisons (Sugihara & Tanaka, 2001; Sugihara, Ishii, & Tanaka, 2004). 

The interval weights are obtained so as to include the given comparisons as close as 

possible. The crisp comparisons are extended into interval ones to reflect our uncertain 

judgments (Saaty & Vargas, 1987; Arbel, 1989). In group AHP, some works handle the 

interval or fuzzy comparisons, instead of crisp comparisons (Dopazo, Chouinard & 

Guisse, 2014; Xu, 2013). However, this study handles the crisp comparisons and the 

proposed approach with crisp comparisons can be extended into the interval comparisons 

on the same principle.  

 



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

428 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

Group decision making is discussed from the viewpoint of AHP (Dyer & Forman, 1992; 

Basak & Saaty, 1993). There are two approaches to aggregate the individuals into a group 

by geometric mean and so on (Aczel & Saaty, 1983; Altuzarra, Moreno-Jimenez, & 

Salvador, 2007; Entani & Inuiguchi, 2010; Forman, Peniwati, 1998; Yeh & Chang, 2009). 

One approach is to aggregate the individual judgments first and then the group decision is 

obtained from the aggregated judgments. The other approach is to aggregate the 

individual decisions which are independently obtained from the individually given 

judgments. Since the latter aggregation can show a decision maker his/her decision, it 

helps him/her to understand the relationship between his/her decision and the group 

decision. This study also assumes that the group decision is obtained as the aggregation 

of individual decisions and discusses how to aggregate them. For a consensus, the group 

weight is obtained so as to have something in common with each individual weight. The 

quality of the group decision is measured by the satisfaction or dissatisfaction of the 

decision makers. Some parts of the group interval weights are supported by a decision 

maker but the other parts are not. Therefore, the satisfaction and dissatisfaction of all 

decision makers are maximized and minimized, respectively. However, they have a trade-

off relationship so that the importance of the satisfaction or dissatisfaction is introduced 

and the group decision depends on it.  

 

2. Individual decisions by Interval AHP 

In AHP, decision maker 𝑘 gives the following pairwise comparison matrix  
 

 𝐴𝑘 = [

1 ⋯ 𝑎𝑘1𝑛
⋮ 𝑎𝑘𝑖𝑗 ⋮

𝑎𝑘𝑛1 ⋯ 1
] , ∀𝑘, (1) 

 

where 𝑎𝑘𝑖𝑗 is his/her intuitive judgment on the importance ratio of alternative 𝑖 to that of 

alternative 𝑗. The comparisons satisfy 𝑎𝑘𝑖𝑖 = 1 and 𝑎𝑘𝑖𝑗 = 1/𝑎𝑘𝑗𝑖  for all 𝑘, 𝑖 and 𝑗. The 

comparisons are consistent if and only if  

 

 
𝑎𝑘𝑖𝑗 = 𝑎𝑘𝑖𝑙 𝑎𝑘𝑙𝑗 , ∀𝑖, 𝑗, 𝑙. (2) 

 

By the eigenvector method, the weights of alternatives are obtained as the eigenvector 

corresponding to principal eigenvalue, 𝐴𝑘 𝒘𝑘 = 𝜆𝒘𝑘 , where 𝒘𝑘 = (𝑤𝑘1, … , 𝑤𝑘𝑛)
𝑇  and 

∑ 𝑤𝑘𝑖𝑖 = 1. It is noted that the weights, 𝑤𝑘𝑖 , ∀𝑖 are crisp.   
 

In Interval AHP, it is assumed that the given comparisons are inconsistent since the 

weights of alternatives are uncertain (Sugihara & Tanaka, 2001; Sugihara et al., 2004). 

An alternative is compared to the other 𝑛 − 1 alternatives. For example, denoting the 
weights of alternative 1 in giving comparisons 𝑎𝑘12  and 𝑎𝑘13  as 𝑤𝑘1

2  and 𝑤𝑘1
3 , 

respectively, they are not always equal, 𝑤𝑘1
2 ≠ 𝑤𝑘1

3 . These weights of alternative 1 

depend on which alterative it is compared to. Then, the interval weight of alternative 1, 

𝑊𝑘1 = [𝑤𝑘1, �̅�𝑘1], includes these weights as 𝑤𝑘1 ≤ 𝑤𝑘1
2 ≤ 𝑤𝑘1 and 𝑤𝑘1 ≤ 𝑤𝑘1

3 ≤ 𝑤𝑘1. 
In this way, the uncertain weight of an alternative is denoted as an interval. The weight of 

alternative 𝑖 is denoted as interval 𝑊𝑘𝑖 = [𝑤𝑘𝑖 , 𝑤𝑘𝑖 ], whose width 𝑤𝑘𝑖 − 𝑤𝑘𝑖  represents 
its uncertainty. In other words, the decision maker 𝑘 uses a real value in the interval 
weight 𝑊𝑘𝑖 in the given comparison 𝑎𝑘𝑖𝑗, where 𝑗 ≠ 𝑖. The problem to obtain the interval 

weights is formulated as the following linear programming (LP) problem.  



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

429 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

 

 

min      ∑ (
𝑖

𝑤𝑘𝑖 − 𝑤𝑘𝑖 ),

s. t. ∑ 𝑤𝑘𝑖
𝑖≠𝑗

+ 𝑤𝑘𝑗 ≥ 1, ∀𝑗,

∑ 𝑤𝑘𝑖
𝑖≠𝑗

+ 𝑤𝑘𝑗 ≤ 1, ∀𝑗,

𝑤𝑘𝑖
𝑤𝑘𝑗

≤ 𝑎𝑘𝑖𝑗 ≤
𝑤𝑘𝑖
𝑤𝑘𝑗

, ∀𝑖, 𝑗,

𝑤𝑘𝑖 ≥ 𝜖, ∀𝑖.

 (3) 

 

The first two constraints are for the normalization of intervals based on interval 

probability (de Campos, Huete, & Moral, 1994; Tanaka, Sugihara, & Maeda, 2004). The 

redundancy of the intervals to make their sum 1 is excluded. For instance, the 1st 

inequality for 𝑗  requires 𝑤𝑘𝑗  to not be too small. When the weights are crisp, 𝑤𝑘𝑖 =

𝑤𝑘𝑖 = 𝑤𝑘𝑖 , ∀𝑖, two inequalities are replaced into ∑ 𝑤𝑘𝑖𝑖 = 1. The next inequalities are the 
inclusion constraints. These inequalities require the obtained interval weights to include 

the given comparisons as  

 

 𝑎𝑘𝑖𝑗 ∈
𝑊𝑘𝑖
𝑊𝑘𝑗

=
[𝑤𝑘𝑖 , 𝑤𝑘𝑖 ]

[𝑤𝑘𝑗 , 𝑤𝑘𝑗 ]
= [

𝑤𝑘𝑖
𝑤𝑘𝑗

,
𝑤𝑘𝑖
𝑤𝑘𝑗

], (4) 

 

where the fraction of intervals is defined as its maximum range. In the case of interval 

comparison, [𝑎𝑘𝑖𝑗 , �̅�𝑘𝑖𝑗 ], it replaces crisp comparison 𝑎𝑘𝑖𝑗 in Equation 4. By minimizing 

the widths of the interval weights, the obtained weights are as close as possible to the 

given comparisons because of the inclusion constraints Equation 4. It is, in other words, 

minimizing uncertainty of the decision. If the comparisons are consistent as in Equation 2, 

the upper and lower bounds of the interval weights are equal, 𝑤𝑘𝑖 = 𝑤𝑘𝑖 , so that the 

weights are crisp as 𝑤𝑘𝑖 = 𝑤𝑘𝑖 = 𝑤𝑘𝑖 , ∀𝑖. They are equal to those by the geometric mean 
and eigenvector methods. The more inconsistent the given comparisons are, the wider the 

obtained interval weights become. The inconsistency of the given comparisons is 

measured by the optimal objective function value of Equation 3.  

 

3. Group decision from individual decisions 
3.1  Relation between group and individual decisions 

In the case of a group of 𝑚  decision makers, there are 𝑚  sets of individual interval 
weights. They are independently obtained from their comparison matrices by Equation 3. 

The group decision is also denoted as interval weights 𝑊𝑖 = [𝑤𝑖 , 𝑤𝑖 ]  and they are 
normalized as  

 

∑ 𝑤𝑖
𝑖≠𝑗

+ 𝑤𝑗 ≥ 1, ∀𝑗,

∑ 𝑤𝑖
𝑖≠𝑗

+ 𝑤𝑗 ≤ 1, ∀𝑗.
 (5) 

 



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

430 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

Then, based on the idea that a decision maker accepts the group decision if it has 

something in common with his/her decision, the group interval weight should satisfy the 

following conditions.  

 

 𝑤𝑖 ≤ 𝑤𝑘𝑖 , 𝑤𝑘𝑖 ≤ 𝑤𝑖  ∀𝑘 ↔ 𝑤𝑖 ≤ min
𝑘

𝑤𝑘𝑖 , max
𝑘

𝑤𝑘𝑖 ≤ 𝑤𝑖 , (6) 

 

where the group decision intersects to all the individual decisions as 𝑊𝑖 ∩ 𝑊𝑘𝑖 ≠ ∅ ∀𝑘. 
In Equation 6, the upper and lower bounds of the group interval weight are approximated 

by the lower and upper bounds of the individual interval weights, respectively. It is based 

on the conjunction approximation. It is noted that the group interval weight complements 

the individual interval weights from the possibility viewpoint.  

 

The group decision includes and is included in the individual decisions as the extreme 

cases of Equation 6. Assuming that the group decision includes all the individual 

decisions as 𝑊𝑖 ⊇ 𝑊𝑘𝑖  ∀𝑘, Equation 6 is rewritten as  
 

 
𝑤𝑖 ≤ 𝑤𝑘𝑖 ≤ 𝑤𝑘𝑖 , 𝑤𝑘𝑖 ≤ 𝑤𝑘𝑖 ≤ 𝑤𝑖 , ∀𝑘 ↔ 𝑤𝑖 ≤ 𝑤𝑘𝑖 ≤ 𝑤𝑘𝑖 ≤ 𝑤𝑖 , ∀𝑘

↔ 𝑤𝑖 = min
𝑘

𝑤𝑘𝑖 , max
𝑘

𝑤𝑘𝑖 = 𝑤𝑖 .
 (7) 

 

Such a group decision is based on the upper approximation of the individual decisions. 

All the possible evaluations in the individual decisions are included in the group decision. 

The group interval weight whose bounds are 𝑤𝑖 < min𝑘 𝑤𝑘𝑖  and max𝑘 𝑤𝑘𝑖 < 𝑤𝑖  is 
redundant, since it includes the weight which is out of any decision maker’s interval 

weight. By Equation 6, some evaluations in an individual decision may be ignored 

compared to the others’ decisions, but the more precise group decision than that is 

obtained by Equation 7. While, based on their lower approximation, the group decision is 

included in all the individual decisions as 𝑊𝑖 ⊆ 𝑊𝑘𝑖 , ∀𝑘, Equation 6 is rewritten as  
 

 
𝑤𝑖 ≤ 𝑤𝑖 ≤ 𝑤𝑘𝑖 , 𝑤𝑘𝑖 ≤ 𝑤𝑖 ≤ 𝑤𝑖 , ∀𝑘 ↔ 𝑤𝑘𝑖 ≤ 𝑤𝑖 ≤ 𝑤𝑖 ≤ 𝑤𝑘𝑖 , ∀𝑘

↔ max
𝑘

𝑤𝑘𝑖 = 𝑤𝑖 , 𝑤𝑖 = min
𝑘

𝑤𝑘𝑖 , if  max
𝑘

𝑤𝑘𝑖 ≤ min
𝑘

𝑤𝑘𝑖 .
 (8) 

 

If max𝑘 𝑤𝑘𝑖 > min𝑘 𝑤𝑘𝑖, there is no group weight included in all the individual weights. 
Equations 6 or 7 result in a group decision for any individual decisions, but Equation 8 

requires the individual decisions have something in common with each other. This study 

defines the general condition of the group decision by Equation 6, and uses Equations 7 

and 8 as its extreme cases. 

 
3.2 Satisfaction and dissatisfaction of decision maker 

By Equation 6 the individual and group decisions intersect each other and the group 

interval weight has something in common with each individual interval weight. A 

decision maker is satisfied with the group decision if s/he can support most of it. Then, 

the similarity of the group decision to his/her decision measures his/her satisfaction. 

While, s/he is not satisfied with the group decision if s/he cannot support some of it. Then, 

the difference of the group decision from his/her decision measures his/her dissatisfaction.   

 

In Figure 1, an individual interval weight is illustrated as the top line and the possible 

group weights which satisfy Equation 6 are illustrated as the following four lines. His/her 



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

431 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

satisfaction and dissatisfaction of an alternative are dented as 𝛼 and 𝛽, respectively. The 
lower two lines show the group weights based on the lower and upper approximations of 

the individual weights, respectively.  

 

 
Figure 1. Individual and group decisions 

 

Decision maker 𝑘  is satisfied with the group weight of alternative 𝑖  with degree 𝛼𝑘𝑖 , 
which is defined as follows.  

 

 𝛼𝑘𝑖 = min { (𝑤𝑖 − 𝑤𝑘𝑖 ), (𝑤𝑘𝑖 − 𝑤𝑖 ), (𝑤𝑖 − 𝑤𝑖 ), (𝑤𝑘𝑖 − 𝑤𝑘𝑖 )}. (9) 
 

This range in the group interval weight is included in the individual one. Since s/he 

supports this range, his/her satisfaction is measured as the sum of these ranges of all 

alternatives, 𝛼𝑘 = ∑ 𝛼𝑘𝑖𝑖 . Then, the sum of the satisfaction of all decision makers should 
be maximized for the better group decision as  

 

 max ∑ 𝛼𝑘
𝑘

= max ∑ 𝛼𝑘𝑖
𝑘𝑖

. (10) 

 

On the other hand, the dissatisfaction degree 𝛽𝑘𝑖 is defined as follows.  
 

 𝛽𝑘𝑖 = max { (𝑤𝑘𝑖 − 𝑤𝑖 ), (𝑤𝑖 − 𝑤𝑘𝑖 ),0, (𝑤𝑘𝑖 − 𝑤𝑖 + 𝑤𝑖 − 𝑤𝑘𝑖 )}. (11) 
 

This range in the group interval weight is not included in the individual one so that it is 

not supported by him/her. Then, the sum of the dissatisfaction with all alternatives should 

be minimized as  

 

 min  ∑ 𝛽𝑘
𝑘

= min  ∑ 𝛽𝑘𝑖
𝑘𝑖

. (12) 

 

As shown in Figure 1, the group interval weight of alternative 𝑖 is divided into the ranges 
satisfying and dissatisfying decision maker 𝑘 as  
 

 𝑤𝑖 − 𝑤𝑖 = 𝛼𝑘𝑖 + 𝛽𝑘𝑖 . (13) 
 

It is a trade-off between the increase of satisfaction and the decrease of dissatisfaction.  

When the group interval weight of an alternative is uncertain so that its width is wide, 

both the satisfaction and dissatisfaction tend to be large.  

 



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

432 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

3.3 Model to obtain group decision 

The group decision is denoted as the normalized interval weights as Equation 5 and 

should have something in common with each individual decision as Equation 6 for a 

consensus. Under these conditions, the satisfaction and dissatisfaction of the decision 

makers are maximized and minimized, respectively, as Equation 10 with Equation 9 and 

Equation 12 with Equation 11. Then, the problem to obtain a group decision is 

formulated as follows.  

 

 

max   ∑ 𝛼𝑘𝑖
𝑘𝑖

,

min   ∑ 𝛽𝑘𝑖
𝑘𝑖

,

s. t. 𝑤𝑖 ≤ 𝑤𝑘𝑖 , ∀𝑖, 𝑘,

𝑤𝑘𝑖 ≤ 𝑤𝑖 , ∀𝑖, 𝑘,

𝑤𝑖 − 𝑤𝑘𝑖 ≥ 𝛼𝑘𝑖 , ∀𝑖, 𝑘,

𝑤𝑘𝑖 − 𝑤𝑖 ≥ 𝛼𝑘𝑖 , ∀𝑖, 𝑘,

𝑤𝑖 − 𝑤𝑖 ≥ 𝛼𝑘𝑖 , ∀𝑖, 𝑘,

𝑤𝑘𝑖 − 𝑤𝑘𝑖 ≥ 𝛼𝑘𝑖 , ∀𝑖, 𝑘,

𝑤𝑘𝑖 − 𝑤𝑖 ≤ 𝛽𝑘𝑖 , ∀𝑖, 𝑘,

𝑤𝑖 − 𝑤𝑘𝑖 ≤ 𝛽𝑘𝑖 , ∀𝑖, 𝑘,

0 ≤ 𝛽𝑘𝑖 , ∀𝑖, 𝑘,

𝑤𝑘𝑖 − 𝑤𝑖 + 𝑤𝑖 − 𝑤𝑘𝑖 ≤ 𝛽𝑘𝑖 , ∀𝑖, 𝑘,

∑ 𝑤𝑖
𝑖≠𝑗

+ 𝑤𝑗 ≥ 1, ∀𝑗,

∑ 𝑤𝑖
𝑖≠𝑗

+ 𝑤𝑗 ≤ 1, ∀𝑗,

𝜖 ≤ 𝑤𝑖 ≤ 𝑤𝑖 , ∀𝑖,

 (14) 

 

where the variables are the upper and lower bounds of the group interval weight, 𝑤𝑖  and 
𝑤𝑖, and the individual satisfaction and dissatisfaction, 𝛼𝑘𝑖 and 𝛽𝑘𝑖. The upper and lower 
bounds of a group interval weight are approximated by the lower and upper bounds of all 

the individual interval weights by Equation 3.   

 

As mentioned in Section 2, the conjunction approximation as the general condition of the 

group decision by Equation 6 includes the extreme conditions as the upper and lower 

approximations by Equations 7 and 8, respectively. Based on the upper approximation as 

𝑊𝑘𝑖 ⊆ 𝑊𝑖 = [min𝑘 𝑤𝑘𝑖 , max𝑘 𝑤𝑘𝑖 ] by Equation 7 illustrated as the lowest line in Figure 
1, decision maker 𝑘 is completely satisfied with the group decision. In this case, his/her 
satisfaction becomes the maximum, 𝛼𝑘𝑖 = 𝑤𝑘𝑖 − 𝑤𝑘𝑖  which is the width of his/her 
interval weight. In order for a group interval weight to include all the individual interval 

weights, the width of the group interval weight tends to be wide, especially when the 

individual weights are not very similar. A part of the group weight supported by an 

individual may not be supported by the other individuals. Therefore, as a result of 

maximizing the satisfaction of all individuals, the dissatisfaction is increased. On the 

other hand, decision maker 𝑘 is not dissatisfied with the group decision when the group 



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

433 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

decision is based on the lower approximation as 𝑊𝑘𝑖 ⊇ 𝑊𝑖 = [max𝑘 𝑤𝑘𝑖 , min𝑘 𝑤𝑘𝑖 ] by 
Equation 8 illustrated as the 2nd line from the lowest in Figure 1. In this case, his/her 

dissatisfaction becomes the minimum, 𝛽𝑘𝑖 = 0. When all the individual interval weights 
are common with each other as max𝑘 𝑤𝑘𝑖 ≤ min𝑘 𝑤𝑖𝑘  as in Equation 8, the 
dissatisfaction of all decision makers is 0. Though, this is seldom satisfied, since the 

individual decisions are often different. If the individual weight is crisp as 𝑤𝑘𝑖 = 𝑤𝑘𝑖 =
𝑤𝑘𝑖 , it can support a point of the group interval weight. His/her satisfaction and 
dissatisfaction are 𝛼𝑘𝑖 = 0 and 𝛽𝑘𝑖 = 𝑤𝑖 − 𝑤𝑖, respectively. Therefore, the ranges of the 
satisfaction and dissatisfaction are 0 ≤ 𝛼𝑘𝑖 ≤ 𝑤𝑘𝑖 − 𝑤𝑘𝑖  and 0 ≤ 𝛽𝑘𝑖 ≤ 𝑤𝑖 − 𝑤𝑖 , 
respectively.   

 

For calculation, two objective functions in Equation 14 are aggregated by the weighting 

approach as  

 max   𝜆 ∑ 𝛼𝑘𝑖
𝑘𝑖

− (1 − 𝜆) ∑ 𝛽𝑘𝑖
𝑘𝑖

, (15) 

 

where 𝜆  and (1 − 𝜆)  are the weights for satisfaction and dissatisfaction, respectively. 
Focusing on the satisfaction with 𝜆 = 1, the group decision is obtained as the upper 
approximation of the individual decisions by Equation 7. While focusing on the 

dissatisfaction with 𝜆 = 0, the group decision approaches the lower approximation of the 
individual decisions by Equation 8.   

 

The inconsistency among the given comparisons by a decision maker is reflected in 

his/her interval weights given by Equation 3. The individual interval weights include all 

the possibilities in his/her judgments. It may be uncomfortable for a decision maker to 

accept the weights out of his/her possible decision. S/he tends to disagree with the group 

decision if it includes the weight which s/he cannot support by his/her possible decision. 

From this viewpoint, it is reasonable to primarily minimize the dissatisfaction and 

secondarily maximize the satisfaction in order to be 1 − 𝜆 > 𝜆.   
 

In Equation 14, the sums of the satisfaction and dissatisfaction with all alternatives of all 

decision makers are maximized and minimized, respectively. Since the satisfaction and 

dissatisfaction of each decision maker are not considered, it happens that the group 

decision which satisfies the specific decision maker may be obtained as the optimal 

solution of LP problem (Equation 14). For the fairness of the decision makers, it is useful 

to determine the appropriate thresholds of the satisfaction and dissatisfaction of a 

decision maker. However, it is not easy, since the satisfaction and dissatisfaction depend 

on the inconsistency in the given comparisons. Instead, it can be done by minimizing the 

deviation of the most and the least satisfaction and dissatisfaction of all the decision 

makers as follows.  

 



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

434 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

 

min   𝜆(𝛼 − 𝛼) + (1 − 𝜆)(𝛽 − 𝛽)

𝑠. 𝑡. Constraints in (14),

𝛽 ≤ ∑ 𝛽𝑘𝑖
𝑖

≤ 𝛽, ∀𝑘,

𝛼 ≤ ∑ 𝛼𝑘𝑖
𝑖

≤ 𝛼, ∀𝑘,

∑ 𝛽𝑘𝑖
𝑖

≤ ∑ 𝛽𝑘𝑖
∗

𝑖

, ∀𝑘,

∑ 𝛼𝑘𝑖
𝑖

≥ ∑ 𝛼𝑘𝑖
∗

𝑖

, ∀𝑘,

 (16) 

 

where 𝛼𝑘𝑖
∗ , ∀𝑘, 𝑖 and 𝛽𝑘𝑖

∗ , ∀𝑘, 𝑖 are the optimal solutions of Equation 14. The sums of the 
satisfaction and dissatisfaction are maximized and minimized primarily in Equation 14 

and under the condition the maximum deviations among the satisfaction and 

dissatisfaction of all decision makers are minimized in Equation 16. As a result, the group 

decision considers the fairness of the individual decision makers when some decision 

maker gives more inconsistent comparisons than the others.  

 

4. Numerical example 

Three decision makers give the following crisp comparison matrices of 4 alternatives 

independently.  

 

 

𝐴1 = [

1 2 4 8
1/2 1 2 4
1/4 1/2 1 2
1/8 1/4 1/2 1

] , 𝐴2 = [

1 3 3 4
1/3 1 3 3
1/3 1/3 1 4
1/4 1/3 1/4 1

] ,

𝐴3 = [

1 1 4 6
1 1 1 2

1/4 1 1 3
1/6 1/2 1/3 1

] .

 

 

(17) 

 

By Equation 3, each decision is obtained as the following interval weights of alternatives.  

 

 

𝑊1 = [

0.533
0.267
0.133
0.067

] , 𝑊2 = [

0.571
[0.190,0.214]
[0.071,0.190]

[0.048,0.143]

] ,

𝑊3 = [

0.390
[0.244,0.390]
[0.098,0.244]

[0.065,0.122]

] .

 (18) 

 

All the decision makers evaluate alternative 1 the best and agree with the weights of 

alternatives 3 and 4 as 0.133 and 0.067, respectively. They are the common weights of all 

individual weights.   



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

435 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

 

In comparison, the crisp weights are obtained by geometric mean method, where the 

weight is obtained as a geometric mean of the comparisons, 𝑤𝑘𝑖 = (∏ 𝑎𝑘𝑖𝑗𝑗 )
1/𝑛

/

∑ (∏ 𝑎𝑘𝑖𝑗𝑗 )
1/𝑛

𝑖 .  

 

 𝑤1 = [

0.533
0.267
0.133
0.067

] , 𝑤2 = [

0.494
0.265
0.165
0.048

] ,   𝑤3 = [

0.467
0.251
0.196
0.086

].  (19) 

 

The comparisons by decision maker 1, 𝐴1, are consistent as in Equation 2 so that his/her 
decision, 𝑊1, is denoted as crisp weights. They are equal to those by geometric mean and 
eigenvector methods. As for the other decision makers, 𝐴2 and 𝐴3, the rough rankings by 
Equations 18 and 19 are not contradicted. When compared to the plausible evaluations by 

the crisp weights in Equation 19, the interval weights by Interval AHP in Equation 18 

show us the possible evaluations. It is possible that alternative 3 in A2 is evaluated as less 

important by the interval weights than by the crisp weights. Similarly in A3, we find that 

alternative 2 may be evaluated as equal to alternative 1. The given comparisons are 

condensed into the crisp weights for the plausible decision. The interval weights give the 

possible decision including the given comparisons. The interval weights are useful 

enough to aid in decision making, since a decision is often at the decision maker’s 

discretion.    

 

Since the individual decision reflects all the possibilities in the given comparisons, it is 

reasonable that the weight for the dissatisfaction, 1 − 𝜆 , is more than that of the 
satisfaction, 𝜆. Then, 𝜆 is supposed to be 0.1 and 0.4, where 1 − 𝜆 > 𝜆, and in addition  
0.9. First, Equation 14 is solved and the satisfaction and dissatisfaction of each decision 
maker are obtained. Then, Equation 16 is solved within them and the group interval 

weights are obtained.   

 

𝑊(𝜆 = 0.1) = [

[0.390,0.571]
[0.214,0.267]
[0.133,0.200]
[0.067,0.143]

]   (width:  0.377),

𝑊(𝜆 = 0.4) = [

[0.390,0.571]
[0.214,0.267]
[0.098,0.200]
[0.066,0.143]

]   (width:  0.413),

𝑊(𝜆 = 0.9) = [

[0.390,0.571]
[0.190,0.390]

[0.071,0.244]
[0.048,0.143]

]   (width:  0.649).

 (20) 

 

In comparison, the group crisp weights as the geometric means of the corresponding three 

individual crisp weights in Equation 19 are as follows.  

 



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

436 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

 𝑤 = [

0.497
0.261
0.163
0.076

] (21) 

 

Since the given comparisons are the same, the group weights by two methods are similar 

and the crisp group weights in Equation 21 are included in the interval group weights in 

Equation 20. The interval group weights show us the possibilities of evaluations from the 

individual decisions. We find that the evaluation of alternative 1 is the most uncertain 

because of its wide width. As for the relationship between alternatives 3 and 4, there is a 

possibility that alternative 4 is evaluated better than alternative 3.  

 

The individual weights of alternative 1 by all decision makers are crisp in Equation 18 so 

that his/her satisfaction cannot be more than 0 and his/her dissatisfaction is equal to the 

width of its group interval weight, 0.181, which does not depend on 𝜆 . When the 
satisfaction is primarily maximized with 𝜆 = 0.9, all the individual interval weights are 
included in the group interval weight with the minimum width. The group decision is 

from their minimum to their maximum, 𝑊𝑖 = [min { 𝑤1𝑖 , 𝑤2𝑖 , 𝑤3𝑖 }, max { �̅�1𝑖 , �̅�2𝑖 , �̅�3𝑖 }]. 
When the importance of the satisfaction, 𝜆, increases from 0.1 to 0.9, the widths of the 
group interval weights become wide from 0.377 to 0.649. The group decision becomes 

uncertain so as to satisfy the decision makers by increasing the supported ranges. From 

the opposite viewpoint of the dissatisfaction, when its importance, 1 − 𝜆, increases from 
0.1 to 0.9, the width of the group interval weight decreases from 0.649 to 0.377 by 

reducing the unsupported ranges. The group interval weights with various importance of 

satisfaction 𝜆s are not very different.  
 

In cases where there are two kinds of the importance of satisfaction and dissatisfaction 

(𝜆, 1 − 𝜆)  as (0.1,0.9)  and (0.4,0.6) , the satisfaction and dissatisfaction of decision 
maker 𝑘, 𝛼𝑘 and 𝛽𝑘, respectively, are compared in Equation 22.  
 

 
(0.1,0.9): (𝛼1, 𝛼2, 𝛼3) = (0,0.133,0.145), (𝛽1, 𝛽2, 𝛽3) = (0.377,0.244,0.232),
(0.4,0.6): (𝛼1, 𝛼2, 𝛼3) = (0,0.169,0.181), (𝛽1, 𝛽2, 𝛽3) = (0.413,0.244,0.232).

 (22) 

 

With (𝜆, 1 − 𝜆) = (0.1,0.9) , the sum of the satisfaction and dissatisfaction of each 
decision maker is 0.377 and that with (𝜆, 1 − 𝜆) = (0.4,0.6) is 0.413. They are equal to 
the sums of the widths of the group interval weights as in Equation 13 and Figure 1. The 

importance of the satisfaction, 𝜆, increases from 0.1 to 0.4, the satisfaction of decision 
makers 2 and 3 also increase by 0.036. Correspondingly, the dissatisfaction of decision 

maker 1, whose satisfaction cannot go over 0, increases by 0.036 since the importance of 

the dissatisfaction, 1 − 𝜆, decreases from 0.9 to 0.6. There is a trade-off between the 
increase of the satisfaction and the decrease of the dissatisfaction.  

 

5. Conclusion 

The individual and group decisions in this study are both denoted as the normalized 

interval weights of alternatives. At first, the individual decisions are independently 

obtained from the crisp comparisons given by the corresponding decision makers. Then, 

the group decision is obtained so as to have something in common with each individual 

decision. The proposed model minimizes the deviations of the upper and lower bounds of 

the group interval weight from those of each individual interval weight. Therefore, the 



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

437 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

obtained group decision is the conjunction approximation of the individual decisions. A 

decision maker supports a part of the group interval weight but s/he does not support the 

other part of it. In other words, the group interval weight is divided into the ranges 

satisfying and dissatisfying a decision maker. For a better group decision, the satisfaction 

and dissatisfaction of all decision makers should be maximized and minimized, 

respectively. However, since these two objectives have a trade-off relationship, they are 

aggregated by the weighting approaches. As the extreme case of the conjunction 

approximation, the group decision can be obtained as the upper or lower approximation 

of the individual decisions. The former group decision includes all the individual 

decisions by primarily maximizing the satisfaction. The latter group decision is included 

in all the individual decisions by primarily minimizing the dissatisfaction. The group 

interval weights reflect the possibilities in the given crisp comparisons and the 

relationship among the individuals. Since they show us the possible evaluations of 

alternatives with the satisfaction or dissatisfaction degree, they are useful in decision 

making.  

 

In this paper, the group interval weights are obtained as the approximation of the 

individual interval weights from the crisp comparison matrices. In future work, the group 

weight approximation process can be combined into determining the individual weights 

from the comparisons. Moreover, the comparisons can be extended into interval and 

fuzzy to be more suitable for our intuitive judgments.  

  



IJAHP Article: Entani/Group interval weights based on conjunction approximation of individual 
interval weights 

 International Journal of the 
Analytic Hierarchy Process 

438 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.277 

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