IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

1 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

 

A NEW SOLUTION FOR INCOMPLETE AHP MODEL USING 

GOAL PROGRAMMING AND SIMILARITY FUNCTION 

 

Maryam Bagheri Fard Sharabiani 

Maryambagherifard4@gmail.com 

School of Industrial Engineering,  

Iran University of Science and Technology 

 

Mohammad Reza Gholamian 

Gholamian@iust.ac.ir 

School of Industrial Engineering,  

Iran University of Science and Technology 

 

Seyed Farid Ghannadpour 

Ghannadpour@iust.ac.ir 

School of Industrial Engineering,  

Iran University of Science and Technology 
 

 

ABSTRACT 

 

The pairwise comparison matrix (PCM) is a crucial element of the Analytic Hierarchy 

Process (AHP). In many cases, the PCM is incomplete and this complicates the decision-

making process. Hence, the present study offers a novel approach for dealing with 

incomplete information in group decision-making. We present a new model of 

incomplete AHP using goal programming (GP) and the similarity function. The 

minimization of this similarity function reduces errors in decision-making. The proposed 

model will be able to estimate the unknown elements in the pairwise comparison matrix 

and calculate the weight vectors obtained from the matrices. Several examples are 

implemented to elaborate on the estimation of unknown elements and weight vectors in 

the proposed model. The results show that the unknown elements have an acceptable 

value with an appropriate consistency rate. 

 

Keywords: Multi-Criteria Decision-Making (MCDM); Analytic Hierarchy Process 

(AHP); Incomplete AHP; Goal Programming (GP); Similarity Function 

 

 

1. Introduction 

In the multi-criteria decision-making (MCDM) process, the decision-maker (DM) 

attempts to use information and their knowledge of the subject under study to employ 

their opinions and professional background to make decisions. Therefore, the information 

available for decision-making is the feedback provided by the decision-makers. When 

this information is not provided or is unavailable, it causes problems in decision-making 

and a failure to achieve managerial goals. Therefore, it is vital to attain efficient methods 

to deal with decision problems in the case of incomplete information. 

 

mailto:Maryambagherifard4@gmail.com
mailto:Gholamian@iust.ac.ir
mailto:Ghannadpour@iust.ac.ir


IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

2 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

The Analytic Hierarchy Process (AHP) is one of the essential tools for multi-criteria 

decision-making (MCDM). This method helps measure the importance of several options 

relative to each other using pairwise comparisons when objective data for decision-

making is not available. Sometimes the data provided by decision-makers are incomplete, 

and there are various reasons for this, including the following (Harker, 1987): 

 
(1) Lack of enough time for decision-making  

(2) Unwillingness to express an opinion 

(3) Uncertainty about the opinion 

 
The application of pairwise comparison matrices (PCM) with complete information is of 

paramount importance when using the AHP technique. Incomplete pairwise comparison 

matrices (iPCMs) disrupt the decision-making process. Due to the significance of this 

matter, numerous studies have been conducted on this topic. The most common and 

classic method in response to this issue is the revised geometric mean (RGM) method, 

which was proposed by Harker (1987) based on the concept of "connecting path". The 

RGM method enables the decision-maker to achieve the priority vector gained from the 

pairwise comparison matrix even if there is incomplete information. Simply put, in this 

approach there is no estimate for the unknown elements in the pairwise matrix, and only 

the weight priority vector can be reached. One of the other impressive studies in this 

domain is a study conducted by Takahashi which proposed “The Two-Step Method”. In 

the two-step method, the unknown elements in the pairwise comparison matrix are 

estimated and then the priority vector obtained from the matrix is calculated. However, in 

many cases, the priority vector gained from this method gives the same priority to options 

that do not have the same priority (Takahashi, 1990). Therefore, multiple studies have 

been conducted in connection with incomplete comparison matrices that attempt to solve 

the listed problems or develop these primary methods. 

 
Some of these studies make decisions with incomplete information by utilizing 

optimization problems, among which the study of Shiraishi et al. (1998) can be noted. In 

this research, polynomials constructed by the eigenvalues of the matrix (λ) were first 

extracted. The results indicated that if the value of the coefficient 𝐶3 = 0 is considered in 
a polynomial with n-3 degree, then the pairwise comparisons matrix will be consistent. 

Since this property is not established for higher-order polynomials, we can reduce the 

inconsistency of matrices by exploiting this feature. Ultimately, the value of unknown 

elements in incomplete matrices can be estimated. This technique is only applicable to 

detect an unknown element and is problematic for a more significant number of unknown 

elements. In another study, an approach to determine the priority of options from the 

incomplete pairwise comparison matrix is provided (Jianqiang, 2006). The Ternary AHP 

was used to make comparisons and obtain the prioritization vector of options. In this 

method, the order of priority of the options is obtained by solving two linear 

programming models. The ultimate priority of options is achieved by comparing the 

ascending and descending order obtained from these two linear programming models. In 

another study, Bozoki et al. (2011) designed a linear system that monitors the matrix 

consistency ratio. Thus, if the optimal value obtained from this model is more than a 

predetermined inconsistency threshold in their proposed model, there is no need to 

continue filling the matrix with unknown elements. Instead, we must determine the 

elements that caused the inconsistencies in the matrix. Moreover, Dopazo and Ruiz-Tagle 

(2011) proposed a method for obtaining the group priority vector in an incomplete 



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

3 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

comparison matrix. They utilized a multi-objective optimization problem (MOP) to 

approximate the priority vector achieved from the pairwise comparison matrix. Then, the 

logarithmic goal programming formulation was adopted to convert the multi-objective 

planning problem to a single-objective optimization problem, and a weighted priority 

vector was obtained. In another study, Benitez et al. (2015) used the AHP, a dynamic 

decision-making model, and added and removed a criterion to the pairwise comparison 

matrix, which turned the static data input mode into a dynamic one and completed the 

incomplete pairwise comparison matrix. In another study, Faramondi et al. (2020)
 

presented an approach to ranking options in the incomplete pairwise comparative matrix 

by extending the logarithmic least squares method in the information scatter mode. 

 
Other studies that investigate decision-making under conditions of incomplete 

information are available. For example, Van Uden et al. (2002) tried to complete the 

AHP pairwise comparison matrix by employing the concept of the geometric mean. The 

estimation of unknown elements was complex for more than two-elements. Nishizawa 

(2005) solved this problem by repeating the geometric mean method. Furthermore, 

employing graph theory is one of the effective methods to deal with the incomplete 

pairwise comparison matrix, which can be observed in numerous studies. Bernroider et 

al. (2010) suggested a method based on the k-walk procedure from graph theory to obtain 

the weighted priority in the incomplete pairwise comparison matrix. In another study, 

Srdjevic et al. (2014) proposed a novel approach to complete incomplete AHP pairwise 

comparison matrices based on the connecting path method. Also, Chen et al. (2015) 

proposed a way to enhance the consistency of matrices. The connecting path method is 

used both to complete incomplete matrices and improve their consistency. Benitez et al. 

(2019) attempted to perform consistent completion of unknown information in the 

incomplete comparison matrix by adopting graph theory. The results from the study of 

graph theory concerning pairwise comparison matrices suggest that if the graph resulting 

from the matrix is unique after completion, the best elements to complete the incomplete 

pairwise comparison matrices are selected. 

 
Uncertainty management conducted by Hua et al. (2008) proposed the priority vector in 

the incomplete pairwise comparison matrix, a combination of the AHP method and the 

Dempster-Shafer theory (DS-AHP). Also, a ranking procedure by incomplete PCMs 

using information entropy and the DS-AHP method was applied by Pan et al. (2014). 

 
The framework defined in the DS-AHP method can perform decision-making in different 

modes, including complete and incomplete, fuzzy, uncertain information, etc. Shiau 

(2012) applied the Dempster-Shafer method in evaluating sustainable transport, and Hu et 

al. (2016) in Train Control Information Systems (TCIS). 

 
Hsu at al. (2011) found that the incomplete information in the pairwise comparison 

matrix could cause problems in the decision-making process. Therefore, fuzzy methods 

can easily express the intuitions of decision-makers compared to classical AHP 

techniques. MCDM with incomplete linguistic preference relations is applied the same 

way. Dong et al. (2015) applied incomplete information in group decision-making based 

on power geometric operators and triangular fuzzy AHP. Moreover, Jandova et al. (2016) 

estimated interval weights in incomplete PCMs based on a weak consistency. 

 
One beneficial and applicable investigation on incomplete pairwise comparison matrices 



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

4 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

is presented by Zhou et al. (2018). In this study, the unknown elements are completed in 

the pairwise comparison matrix based on the decision-making trial and evaluation 

laboratory (DEMATEL) method. This algorithm transforms the direct-relation matrix to 

the total-relation by the DEMATEL technique and can estimate unknown elements with a 

good consistency ratio. The application of neural networks to make decisions on 

incomplete information is discussed. Hu and Tsai (2006) designed an algorithm with the 

aid of a neural network with a backpropagation approach. The input for this algorithm is 

an incomplete pairwise comparison matrix, and its output is a complete pairwise 

comparison matrix. One of the weaknesses expressed in this model is that we can only 

identify an unknown element of the incomplete pairwise comparison matrix in the 

proposed model. Thus, in another study, Gomez-Ruiz et al. (2010) estimated unknown 

values in the pairwise comparison matrix by designing an algorithm based on the most 

famous model of an artificial neural network named multi-layer perceptron. One of the 

critical features of this algorithm is that it is not merely limited to identifying an unknown 

element and can improve the matrix consistency while completing the pairwise 

comparison matrix.  

 

According to the latest applied AHP studies, a heuristic method to rate the alternatives in 

the AHP was applied in Lin and Kou (2020). A new parsimonious AHP methodology is 

introduced in Abastante et al. (2019). In other studies, Ruiz et al. (2020) proposed the 

GIS-AHP method, Amenta et al. (2021) dealt with the issue of aggregating judgments in 

non-negotiable AHP, and Maleki et al. (2020) proposed a combination method of 

incomplete AHP and Choquet integral. 

 
Based on the studies reviewed, we conclude that most studies conducted in this area have 

dealt with the direct determination of the priority vector to accelerate the achievement of 

the vital goal of decision-making, which is the selection of choices and determination of 

their priority. Simultaneous determination of weight vectors and unknown elements in the 

pairwise comparison matrix has not been well-studied. Therefore, the current study 

presents a new approach to completing the pairwise comparison matrix in the Analytic 

Hierarchy Process (AHP). This study aims to calculate the weight vector gained from 

incomplete pairwise comparison matrices in the AHP by employing the similarity 

function and goal programming (GP) techniques. Additionally, we will strive to identify 

the unknown elements in the incomplete pairwise comparison matrix with the intended 

model. 

 
The GP technique was inspired by Dopazo and Ruiz-Tagle (2011) and the total weights 

estimated in this study. The main difference between the proposed study from previous 

studies is that the unknown elements are estimated here. Also, the weight vector is 

calculated in each group separately and the total weight is estimated by considering the 

impact coefficient. The results of this study have been compared to the results of Zhou et 

al. (2018).  

 

 

2. Preliminaries 

Definition 1. A matrix M is called a pairwise comparison matrix if the condition 𝑎𝑖𝑗 = 

1 𝑎𝑗𝑖⁄  for all, j (Saaty, 1998). 
 



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

5 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

Definition 2. If 𝑀𝑛×𝑛 is a pairwise comparison matrix, 𝜆𝑚𝑎𝑥 is the maximum eigenvalue 
of this matrix, and I is an identity matrix; the weight vector (W) can be calculated using 

the following equation (Saaty, 1998).  

 
(1)    (𝑀 − 𝜆𝑚𝑎𝑥  ×  𝐼) 𝑊 =  0             𝜆𝑚𝑎𝑥≥ n 

 

Definition 3. To calculate the consistency ratio (CR) in the pairwise matrix 𝑀𝑛×𝑛, the 
consistency index must be first calculated as follows: 

 
(2)    𝐶. 𝐼. =      

𝜆𝑚𝑎𝑥−𝑛

𝑛−1
 

 

Thus, the consistency ratio is as follows: 

 
     (3) 

𝐶. 𝑅. =
 𝐶. 𝐼.

𝑅. 𝐼.
 

 

In Equation (3), the random index R.I. is obtained from Table 1. If the consistency ratio is 

less than 0.1, the matrix is consistent, and the resulting weights are acceptable. 

Otherwise, the decision-makers must correct the pairwise comparison matrix (Saaty, 

1998). 

 

Table 1  

Random index (Saaty, 1998) 

 

𝑵 1 2 3 4 5 6 7 8 9 10 
𝑹𝑰 0 0 0.52 0.89 1.12 1.26 1.36 1.41 1.46 1.49 
 

Definition 4. The structure of the incomplete PCMs is as below, where the * indicates 

unknown elements (Harker, 1987). 

 

 𝑴=[

1 ∗ 𝑎13
∗ 1 𝑎23

𝑎31 𝑎32 1
] 

 

 

Definition 5. Group decision-making (GDM) can be defined by a finite set of alternatives 

X = {𝑥1, 𝑥2, . . ., 𝑥𝑛} and a group of experts {𝐸1,..., 𝐸𝑚} in which each expert sorts 
alternatives through a set of criteria C = {𝑐1, 𝑐2, . . ., 𝑐𝑙} (Dopazo & Ruiz-Tagle, 2011) . 
 

 

3. Proposed model 

In this section, the proposed model is elaborated. Therefore, the symbols used in this 

section are presented in the Table 2. 

 
  



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

6 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

Table 2  

Introduction of symbols 

 

Definition Parameter Definition Variable 

Index of decision-

making groups 

k Unknown pairwise matrix 

elements in each decision-

making group  

m𝑖𝑗
𝑘  

Impact coefficient of 

each decision-making 

group  

𝛼𝑘 Weight of each matrix in each 
decision-making group   

 

𝑤𝑖
𝑘 

Number of pairwise 

matrix elements in 

each decision-making 

group  

 

𝐵𝑘 
 

Final weight  

 

𝑤𝑖 

 

Coefficient (weight) 

of objective functions  

 

𝜆𝑓 
Estimation error of pairwise 

matrix elements in each 

decision-making group  

𝑡𝑘 
 

  

 

Maximum error  D 

 

 

In this study, group decision-making was performed, in which the symbol k with the 

value of k=1, 2,…, m  is defined for introducing the groups, and the coefficient 𝛼𝑘 is 
considered as the impact factor of each group. The introduced impact factor is based on 

the importance of the groups, and the preference of the opinions presented will have 

different weight values in the interval of 0 ˂ 𝛼𝑘 ˂ 1 and ∑ 𝛼𝑘
𝑚
𝑘=1 = 1. Therefore, it is 

evident that if opinions are in a group with more importance, the value of the coefficient 

will be closer to one, and if is a group has less importance, this value will be closer to 0. 

The pairwise comparison matrices are presented to the decision-making groups, and the 

unknown values will be estimated by the developed model in each group. Therefore, 

there are two types of weights in this study; one is the weight of each matrix in each 

decision-making group (𝑤𝑖
𝑘), and the other is the overall weight (𝑤𝑖), which is achieved 

from the weighted average of the groups. Concerning the listed cases, a function to 

minimize the error value of decision-making is introduced called the similarity 

function 𝑓𝑘(𝑤).  
 

(4) 
Sim (𝑚𝑘,𝑤𝑘) =  

1

𝐵𝑘
 ∑ |𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘| = 𝑓𝑘(𝑤)𝑖,𝑗  

 

The first and second objective functions to minimize this expression are as follows 

(Dopazo & Ruiz-Tagle, 2011): 

 
    (5) Min∑ 𝛼𝑘. 𝑓𝑘(𝑤)

𝑚
𝑘=1  

(6) Min 𝑚𝑎𝑥𝑘=1,…,𝑚 𝛼𝑘. 𝑓𝑘(𝑤) 

 

Equations (5) and (6) are associated with the values 𝑓1(𝑤), 𝑓2(𝑤), … , 𝑓𝑘(𝑤); these 
functions indicate similarity functions in the first to k-th decision-making groups. 

Therefore, the first objective function ∑ 𝛼𝑘. 𝑓𝑘(𝑤)
𝑚
𝑘=1  represents the weighted sum of 

deviations or errors obtained in the decision groups. It is clear that the lowering of this 



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

7 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

value reduces errors, and a more accurate estimation of problem variables is achieved. 

Also, using the weighted sum of errors causes errors in less important decision-making 

groups to have less impact on the total value. 

 

In the second objective function, 𝑚𝑎𝑥𝑘=1,…,𝑚 𝛼𝑘. 𝑓𝑘(𝑤) indicates the maximum 
deviation. Therefore, the maximum possible error value is minimized by considering the 

whole system's performance. The two objective functions introduced by Equations (5) 

and (6) will be minimized. 

 
Moreover, for the consistent estimation of unknown elements in each group, an error 

value called t
k
 is considered. Therefore, regarding the consistency ratio, we will use the 

following equations to consider the error t
k
: 

 
  (7) 𝑚𝑖𝑗

𝑘 . 𝑚𝑗𝑠
𝑘

𝑚𝑖𝑠
𝑘  ≤ 𝑡

𝑘 

  (8) 

 

𝑚𝑖𝑠
𝑘

𝑚𝑖𝑗
𝑘 𝑚𝑗𝑠

𝑘 ≤ 𝑡
𝑘 

 

By minimizing this error, due to the elements of each matrix, the consistency ratio 

between the elements can be minimized. Thus, the unknown elements can be estimated 

with a lower consistency ratio in each group. Hence, to minimize 𝑡𝑘, another objective 
function is as follows: 

 
(9) Min∑ 𝑡𝑘𝑚𝑘=1  

 

Thus, the overall objective function of the problem is composed of three goals. These 

objectives will be weighted by considering the coefficients 𝜆𝑓 and will be considered as 

an overall objective. In this case, given the number of the listed objectives, the sum of the 

coefficients 𝜆𝑓 will be as 𝜆𝑓1 + 𝜆𝑓2 + 𝜆𝑓3 = 1. 

 

Subsequently, the presented planning will be modelled as goal programming (GP). The 

purpose of planning problems with goal programming is to attain the goal intended in the 

study. In the case of establishing this goal, we will obtain the best solution to the 

problem; otherwise, positive deviation and negative deviation from the goal are taken 

into account. 

 

If 𝑛𝑖𝑗
𝑘  and 𝑝𝑖𝑗

𝑘  are a negative deviation from the goal and a positive deviation from the 

goal by Equations (10) and (11), 

 
(10) 

𝑛𝑖𝑗
𝑘 = 

1

2
 [|𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘| + (𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘)]     (negative) 

     

(11) 
𝑝𝑖𝑗

𝑘 = 
1

2
 [|𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘| − (𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘)]     (positive)  

 

respectively, by placing Equation (4) in the objective function and considering the 

maximum error D, we will have: 

 



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

8 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

(12) 
D = 𝑚𝑎𝑥𝑘=1,…,𝑚   

𝛼𝑘

𝐵𝑘
 ∑ |𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘|𝑖,𝑗    

 

Accordingly, it can be concluded: 

 
(13) 

  
𝛼𝑘

𝐵𝑘
 ∑ |𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘|𝑖,𝑗   D           k =1,2,…m                                 

 

Moreover, considering the values of positive deviation (𝑝𝑖𝑗
𝑘 ) and negative deviation (𝑛𝑖𝑗

𝑘 ), 

the following statement will be obtained. 

 

Corollary 3.1:  For each pairwise comparison matrices in the goal programming model, 

we will have: 

 
       (14)   

|𝑚𝑖𝑗
𝑘 −

𝑤𝑖
𝑘

𝑤𝑗
𝑘|   = 𝑛𝑖𝑗

𝑘 + 𝑝𝑖𝑗
𝑘  

 

Proof 

𝑛𝑖𝑗
𝑘 + 𝑝𝑖𝑗

𝑘  = 
1

2
 [|𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘| + (𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘)] + 

1

2
 [|𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘| − (𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘)] = 

1

2
|𝑚𝑖𝑗

𝑘 −

𝑤𝑖
𝑘

𝑤𝑗
𝑘| + 

1

2
(𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘) +  

1

2
|𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘| - 

1

2
(𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘) = |𝑚𝑖𝑗

𝑘 −
𝑤𝑖

𝑘

𝑤𝑗
𝑘|                                           

□ 

 

Accordingly, the following constraint will be added to the model according to corollary 

3.1. 

 
𝛼𝑘

𝐵𝑘
∑ (𝑛𝑖𝑗

𝑘 + 𝑝𝑖𝑗
𝑘 ) ≤  D𝑖,𝑗            k =1,2,…m             (15)                                                                  

 

Thus, the transformation of the proposed model into goal programming is as follows: 

 
 (16) min {(1 − (𝜆𝑓1 + 𝜆𝑓2)) ∑ .

𝛼𝑘

𝐵𝑘
∑ |𝑛𝑖𝑗

𝑘 + 𝑝𝑖𝑗
𝑘 | + 𝜆𝑓1. 𝐷𝑖,𝑗 + 𝜆𝑓2. ∑ 𝑡

𝑘𝑚
𝑘=1

𝑚
𝑘=1 } 

 S.t 

 
𝛼𝑘
𝐵𝑘

∑(𝑛𝑖𝑗
𝑘 + 𝑝𝑖𝑗

𝑘 ) ≤  D

𝑖,𝑗

  k=1,2,…,m 
 

  (17) 

𝑚𝑖𝑗
𝑘 −

𝑤𝑖
𝑘

𝑤𝑗
𝑘 - 𝑛𝑖𝑗

𝑘 + 𝑝𝑖𝑗
𝑘 = 0 i=1,2,…,n     j=1,2,…,n     k=1,2,…,m   (18) 

 

∑ 𝑤𝑖
𝑘𝑛

𝑖=1 =1 
k=1,2,…,m                                      (19) 

𝑚𝑖𝑗
𝑘 .𝑚𝑗𝑠

𝑘

𝑚𝑖𝑠
𝑘   ≤ 𝑡

𝑘 
1˂i˂j˂s≤n      k=1,2,…,m                                     

    .         
  (20) 

𝑚𝑗𝑠
𝑘

𝑚𝑖𝑗
𝑘 𝑚𝑗𝑠

𝑘  ≤ 𝑡
𝑘 1˂i˂j˂s≤n      k=1,2,…,m                                      (21) 



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

9 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

𝑤𝑖 = ∑ 𝛼𝑘. 𝑤𝑖
𝑘

𝑚

𝑘=1
 i=1,2,…,n       (22) 

∑ 𝑤𝑖
𝑛
𝑖=1 =1     (23) 

𝑡𝑘˃0 , 𝑚𝑖𝑗
𝑘 ˃0 , 𝑊𝑖˃0 , 𝐷˃0 , 𝑤𝑖

𝑘˃0 ,  𝑛𝑖𝑗
𝑘 ≥ 0 , 𝑝𝑖𝑗

𝑘 ≥ 0   (24) 

 

Equation (18) is the goal constraint, aiming to achieve a zero-error rate of decision-

making. 

 

 

4. Numerical example 

In this section, examples of incomplete pairwise comparison matrices in different 

dimensions, taken from the study of Zhou et al. (2018) are solved. In the mentioned 

study, iPCMs were introduced in two categories including with a full consistency ratio 

and without a full consistency ratio (as shown in the Appendix). Then, using the 

DEMATEL method, a new estimation for unknown values of iPCMs is proposed. This 

study was used in our work in such a way that we considered these categories as the 

responses of two groups of decision makers in solving the problem, and therefore, the 

results of Zhou et al. (2018) can be considered as a comparison benchmark to the results 

of this study. Considering 𝜆𝑓1, 𝜆𝑓2 = 0.1 and 𝛼1, 𝛼2 = 0.5, the results are reported in 

Tables 3-7. 

 

Many questions have been raised to validate the proposed model: 

 Which matrices can be solved by the proposed model? 

 What is the effect of the proposed model on matrices with perfect consistency 
(CR = 0) and incomplete consistency? 

 Will the final weight priority vector provide the decision-maker with different 
and acceptable weights? 

 

To respond to the above questions, the following two steps were taken: 

 

Step 1: The proposed model is employed to estimate the unknown elements in 

incomplete comparison matrices. 

 

To this end, several matrices are placed into two decision-making groups. The first group 

of matrices has a consistency ratio, and the second group has matrices with a perfect 

consistency ratio (CR = 0). The unknown elements are estimated, and the results are 

reported in the top two rows of Tables 3-7. 

 
To evaluate the consistency ratio, the same value is considered for the impact factor in 

each group. Also, the matrix dimension and the number of unknown elements increased. 

The missing values are estimated by the proposed model and compared with the 

DEMATEL as a benchmark method. 

 
Step 2: In this step, the iPCMs completed by the DEMATEL and the weight priority 

vector were calculated using the eigenvector method. Then, the weight vector obtained 

from the developed model is compared with the eigenvector as a benchmark method and 

the results are reported in the following sections. 



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

10 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

 
4.1. Matrix collection  

In this section, the iPCMs in the study of Zhou et al. (2018) are exploited for validation. 

These iPCMs have different dimensions which are presented in two groups.  

 
4.2. Validation result and discussion 

The results gained from the first and second validation stages are reported in Tables 3-7. 

 
Table 3  

Validation results of 4 × 4 matrix 

 

m23=1.93   m24=1.25  𝒎𝟑𝟐=0.516  
𝒎𝟑𝟒=0.333      𝒎𝟒𝟐=0.8        𝒎𝟒𝟑=1.55 
 

Group 1  
Estimation of unknown 

elements with research model 

𝒎𝟏𝟐=2       𝒎𝟏𝟒=8    𝒎𝟐𝟏=0.5       
𝒎𝟐𝟒=4               𝒎𝟒𝟏=0.125       
𝒎𝟒𝟐=0.25 

Group 2 

𝒎𝟐𝟑=2   𝒎𝟐𝟒=1.25  𝒎𝟑𝟐=0.52  
𝒎𝟑𝟒=0.65        𝒎𝟒𝟐=0.8           𝒎𝟒𝟑=1.54 
 

Group 1  

Estimation of unknown 

elements with DEMATEL 

model 𝒎𝟏𝟐=2       𝒎𝟏𝟒=8            𝒎𝟐𝟏=0.5       
𝒎𝟐𝟒=4  
𝒎𝟒𝟏=0.13     𝒎𝟒𝟐=0.25 
 

Group2 

CR = 0 Group 1 Consistency Ratio 

CR = 0.001 Group 2 

 

W = [0.395  0.294  0.149  0.162] 

 

_ Calculation of weight priority 

vector with the research model 

 

W = [0.393  0.310  0.158  0.139] 

 

_ Calculation of weight priority 

vector by eigenvector method 

and complete DEMATEL 

matrix 

 

 

  



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

11 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

Table 4  

Validation results of 5 × 5 matrix 

 

𝒎𝟏𝟐=0.889    𝒎𝟏𝟑= 1.600    𝒎𝟐𝟏=0.5 
𝒎𝟐𝟓= 9        𝒎𝟑𝟏=0.625       𝒎𝟑𝟒=1.69             
𝒎𝟒𝟑=0.36       𝒎𝟓𝟐=0.11 
 

Group 1  
Estimation of unknown 

elements with research model 

𝒎𝟐𝟑=1          𝒎𝟐𝟓=4         𝒎𝟑𝟐=1     
𝒎𝟑𝟓=4         𝒎𝟒𝟓=2         𝒎𝟓𝟐=0.25    
𝒎𝟓𝟑=0.25    𝒎𝟓𝟒=0.5 

Group 2 

𝒎𝟏𝟐=0.889 𝒎𝟏𝟑= 1.69 𝒎𝟐𝟏=1.125  𝒎𝟐𝟓=9     
𝒎𝟑𝟏=0.625    𝒎𝟑𝟒=2.77         𝒎𝟒𝟑=0.36  
     𝒎𝟓𝟐=0.11 
 

Group 1 Estimation of unknown 

elements with DEMATEL 

model 

𝒎𝟐𝟑=1            𝒎𝟐𝟓=4        𝒎𝟑𝟐=1               
   𝒎𝟑𝟓=4            𝒎𝟒𝟓=2       𝒎𝟓𝟐=0.25         
     𝒎𝟓𝟑=0.25      𝒎𝟓𝟒=0.5 

 

Group2 

CR=0.014 Group 1  

Consistency Ratio CR=0.002 Group 2 

 

W= [0.372  0.287  0.206  0.089  0.046] 

_ Calculation of weight priority 

vector with the research model 

 

W= [0.375  0.292  0.196  0.091  0.044] 

_ Calculation of weight priority 

vector by eigenvector method 

and complete DEMATEL 

matrix 

 

 

  



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

12 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

Table 5  

Validation results of 6 × 6 matrix 

 

𝒎𝟏𝟑=3.02    𝒎𝟏𝟔= 1.27   𝒎𝟐𝟒=0.39  
𝒎𝟐𝟔=0.26    𝒎𝟑𝟏=0.33      𝒎𝟑𝟓=3.38    
𝒎𝟒𝟐=2.56  𝒎𝟒𝟔=0.80  𝒎𝟓𝟑=0.3          
𝒎𝟔𝟏=0.79           𝒎𝟔𝟐=3.87           𝒎𝟔𝟒=1.25 

Group 1  
Estimation of unknown 

elements with research model 

𝒎𝟏𝟐=1.502          𝒎𝟏𝟒= 3.38    𝒎𝟐𝟏=0.67  
𝒎𝟐𝟒=2.25   𝒎𝟑𝟒=1.5     𝒎𝟒𝟏=0.30     
𝒎𝟒𝟐=0.44 𝒎𝟒𝟑=0.6 𝒎𝟒𝟓=1.5        
𝒎𝟓𝟒=0.67             𝒎𝟔𝟓=0.67 

Group 2 

𝒎𝟏𝟑=3.02      𝒎𝟏𝟔= 1.27     𝒎𝟐𝟒=0.39  
𝒎𝟐𝟔=0.26   𝒎𝟑𝟏=0.33   𝒎𝟑𝟓=3.38   
𝒎𝟒𝟐=2.56  𝒎𝟒𝟔=0.8   𝒎𝟓𝟑=0.3             
𝒎𝟔𝟏=0.79          𝒎𝟔𝟐=3.87          𝒎𝟔𝟒=1.25 

Group 1  

Estimation of unknown 

elements with DEMATEL 

model 

𝒎𝟏𝟐=1.5             𝒎𝟏𝟒= 3.38         𝒎𝟐𝟏=0.67    
𝒎𝟐𝟒=2.25   𝒎𝟑𝟒=1.5       𝒎𝟒𝟏=0.30      
𝒎𝟒𝟐=0.44   𝒎𝟒𝟑=0.67   𝒎𝟒𝟓=1.5        
𝒎𝟓𝟒=0.67           𝒎𝟔𝟓=0.67  

Group2 

CR= 0.048 Group 1  

Consistency Ratio CR= 0.001 Group 2 

 

W= [0.339  0.153  0.133  0.158  0.062  

0.155] 

 

_ 

 

Calculation of weight priority 

vector with the research model 

 

W= [0.380  0.139  0.1494  0.1497  0.057  

0.119] 

_ Calculation of weight priority 

vector by eigenvector method 

and complete DEMATEL 

matrix 

 

  



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

13 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

Table 6  

Validation results of 7 × 7 matrix 

 

𝒎𝟏𝟒=0.259   𝒎𝟏𝟕= 0.778  𝒎𝟐𝟑=6.11   
𝒎𝟐𝟓=0.873    𝒎𝟐𝟕=2.037   𝒎𝟑𝟐=0.164  
𝒎𝟑𝟔=0.214  𝒎𝟒𝟏=4.13  𝒎𝟒𝟓=1.286  
𝒎𝟓𝟐=2.39  𝒎𝟓𝟒=0.99  𝒎𝟔𝟑=5.62  
𝒎𝟔𝟕=1.556   𝒎𝟕𝟏=1.286     𝒎𝟕𝟐=0.491  
𝒎𝟕𝟔=0.643 

Group 1  
Estimation of unknown 

elements with research model 

𝒎𝟏𝟐                 𝒎𝟏𝟑= 1         𝒎𝟏𝟒=2              
𝒎𝟏𝟔=8             𝒎𝟏𝟕=8           𝒎𝟐𝟏=1   
𝒎𝟐𝟑=1            𝒎𝟐𝟒=2          𝒎𝟐𝟓=4     
𝒎𝟑𝟏=1            𝒎𝟑𝟐=1          𝒎𝟒𝟏=0.5 
𝒎𝟒𝟐=0.5            𝒎𝟓𝟐=0.25    
𝒎𝟔𝟏=0.125         𝒎𝟕𝟏=0.125   

Group 2 

𝒎𝟏𝟒=0.24    𝒎𝟏𝟕= 1.2    𝒎𝟐𝟑=6.11  
𝒎𝟐𝟓=0.42  𝒎𝟐𝟕=1.10  𝒎𝟑𝟐=0.16   
𝒎𝟑𝟔=0.18  𝒎𝟒𝟏=4.13  𝒎𝟒𝟓=1.01  
𝒎𝟓𝟐=2.39  𝒎𝟓𝟒=0.99  𝒎𝟔𝟑=5.62  
𝒎𝟔𝟕=1.59          𝒎𝟕𝟏=0.83          𝒎𝟕𝟐=0.91      
𝒎𝟕𝟔=0.63 

Group 1  

Estimation of unknown 

elements with DEMATEL 

model 

𝒎𝟏𝟐=1              𝒎𝟏𝟑= 1               𝒎𝟏𝟒=2  
 𝒎𝟏𝟔=8             𝒎𝟏𝟕=8                𝒎𝟐𝟏=1   
 𝒎𝟐𝟑=1             𝒎𝟐𝟒=2                𝒎𝟐𝟓=4      
 𝒎𝟑𝟏=1             𝒎𝟑𝟐=1                𝒎𝟒𝟏=0.5  
 𝒎𝟒𝟐=0.5      𝒎𝟓𝟐=0.25     𝒎𝟔𝟏=0.125      
𝒎𝟕𝟏=0.125                  

Group 2 

CR= 0.072 Group 1  

Consistency Ratio CR= 0 Group 2 

  

 

W= [0.160  0.217  0.140  0.198  0.137 

0.086  0.061] 

- Calculation of weight priority 

vector with the research model 

 

W= [0.168  0.208  0.146  0.224  0.098  

0.090  0.063] 

- Calculation of weight priority 

vector by eigenvector method 

and complete DEMATEL 

matrix 

 

  



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

14 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

Table 7  

Validation results of 8 × 8 matrix 

 

𝒎𝟏𝟐=2                  𝒎𝟏𝟑= 2            𝒎𝟏𝟕=0.667 
𝒎𝟏𝟖=0.75              𝒎𝟐𝟏=0.5           𝒎𝟐𝟓=2.67  
𝒎𝟐𝟕=0.33              𝒎𝟑𝟏=0.52        𝒎𝟐𝟑=1      
𝒎𝟑𝟐=0.93              𝒎𝟑𝟖=0.38         𝒎𝟒𝟔=1  
𝒎𝟓𝟐=0.33      𝒎𝟓𝟕=0.11         𝒎𝟔𝟒=1                               
𝒎𝟕𝟏=1.5               𝒎𝟕𝟐=3             𝒎𝟕𝟓=9  
𝒎𝟕𝟖=1.125            𝒎𝟖𝟏=1.3            𝒎𝟖𝟑=2.667 
𝒎𝟖𝟕=0.889 

Group 1  

 

 

Estimation of unknown 

elements with research 

model 

𝒎𝟏𝟓=0.5               𝒎𝟏𝟖= 2            𝒎𝟐𝟑=0.25  
  𝒎𝟐𝟕=0.25           𝒎𝟐𝟖=1             𝒎𝟑𝟐=4  
 𝒎𝟑𝟓=1                 𝒎𝟑𝟖=4             𝒎𝟒𝟔=1  
 𝒎𝟒𝟕=0.25            𝒎𝟓𝟏=2             𝒎𝟓𝟑=1  
 𝒎𝟔𝟒=1                 𝒎𝟔𝟖=1             𝒎𝟕𝟐=4   
 𝒎𝟕𝟒=4                 𝒎𝟕𝟖=4             𝒎𝟖𝟏=0.5    
 𝒎𝟖𝟐=1               𝒎𝟖𝟑=0.25                            
𝒎𝟖𝟔=1                  𝒎𝟖𝟕=0.25 

Group 2 

𝒎𝟏𝟐=1.78             𝒎𝟏𝟑= 1.93        𝒎𝟏𝟕=0.69  
𝒎𝟏𝟖=0.81             𝒎𝟐𝟏=0.56         𝒎𝟐𝟑=1.08 
𝒎𝟐𝟓=2.67  𝒎𝟐𝟕=0.40  𝒎𝟑𝟏=0.52                 
𝒎𝟑𝟐=0.93             𝒎𝟑𝟖=0.42         𝒎𝟒𝟔=0.81  
𝒎𝟓𝟐=0.37             𝒎𝟓𝟕=0.1 1         𝒎𝟔𝟒=1.23 
𝒎𝟕𝟏=1.45            𝒎𝟕𝟐=2.52          𝒎𝟕𝟓=9    
𝒎𝟕𝟖=1.29              𝒎𝟖𝟏=1.23  
𝒎𝟖𝟑=2.37             𝒎𝟖𝟕=0.78 

Group 1  

 

Estimation of unknown 

elements with DEMATEL 

model 

𝒎𝟏𝟓=0.5          𝒎𝟏𝟖= 2      𝒎𝟐𝟑 = 𝟎. 𝟐𝟓     
𝒎𝟐𝟕=0.25        𝒎𝟐𝟖=1       𝒎𝟑𝟐=4      
𝒎𝟑𝟓=1             𝒎𝟑𝟖=4       𝒎𝟒𝟔=1      
 𝒎𝟒𝟕 = 𝟎. 𝟐𝟓     𝒎𝟓𝟏=2      𝒎𝟓𝟑=1     
𝒎𝟔𝟒=1              𝒎𝟔𝟖=1       𝒎𝟕𝟐=4   
𝒎𝟕𝟒=4              𝒎𝟕𝟖=4      𝒎𝟖𝟏=0.5  
𝒎𝟖𝟐=1              𝒎𝟖𝟑=0.25 𝒎𝟖𝟔=1  𝒎𝟖𝟕=0.25 

Group 2 

CR= 0.021 Group 1  

Consistency Ratio CR= 0 Group 2 

 

W= [0.149  0.075    0.158    0.043   0.127  0.044  

0.252  0.153] 

_ Calculation of weight 

priority vector with the 

research model 

 

W= [0.159  0.079  0.156  0.042  0.091  0.047  

0.287  0.135] 

_ Calculation of weight 

priority vector by 

eigenvector method and 

complete DEMATEL 

matrix 

 

We infer that the proposed model can be employed in all pairwise comparison matrices 

from the results achieved from this validation. The elements estimated by the estimation 

model provide a desirable consistency ratio, which means we can trust the results and 



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

15 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

accept the obtained priority vector. 

  

Moreover, the elements estimated by the research model were equal to the ones obtained 

by the DEMATEL technique, in many cases representing an acceptable estimation of 

unknown data. The consistency ratio (CR) in a complete PCM with estimated data is less 

than 0.1 and is considered a good consistency ratio. The priority vector gained from the 

model does not have equal or inverse weights relative to the vector obtained by the 

eigenvector method. Hence, we can rely on the results of the proposed model. 

 
In matrices whose consistency ratio is 0 (CR = 0), the estimation of the unknown 

elements by the proposed model is precisely equal to that of the DEMATEL technique. 

Therefore, when the consistency ratio for a matrix is 0, the value of the decision error will 

be 0, and the estimation of unknown elements will be accurate and unique. 

 
4.3. Verification results and discussion 

In order to verify the proposed model, the following question will be answered: How will 

the matrices' consistency ratio vary with an increasing number of unknown elements in a 

fixed dimension? 

 
In this section, by fixing the dimensions of the matrices we deal with increasing the 

number of unknown elements and their relationship with the consistency ratio. The 

matrices employed in this section are the matrices used in Zhou et al. (2018), which were 

examined in different dimensions. The row corresponding to the unknown element pair 

represents the number of unknown elements in the pairwise matrix. (For example, if an 

element 𝑚12 is unknown, its inverse element 𝑚21 is also unknown). 
 

The first group includes matrices with a consistency ratio less than 0.1, and the second 

group comprises matrices with a 0-consistency ratio. Figures 2-6 illustrate the 

relationships between the number of unknown elements and their related consistency 

ratio (CR). Microsoft Excel was used to illustrate these diagrams. 

 

 
 

Figure 2 Variations of CR in 4 × 4 matrix 

 

0

0.02

0.04

0.06

1 2 3

C
R

 

Unknown Pairs 

Variations of CR 

Group 1

Group 2



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

16 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

 
 

Figure 3 Variations of CR in 5 × 5 matrix 

 

 
 

Figure 4 Variations of CR in 6 × 6 matrix 

 

 
 

Figure 5 Variations of CR in 7 × 7 matrix 

 

0

0.02

0.04

0.06

0.08

1 2 3 4

C
R

 

Unknown Pairs 

Variation of CR 

Group 1

Group 2

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 2 3 4 5 6

C
R

 

Unknown Pairs 

Variation of CR 

Group 1

Group 2

0

0.01

0.02

0.03

0.04

0.05

1 2 3 4 5 6 7 8

C
R

 

Unknown Pairs 

Variation of CR 

Group 1

Group 2



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

17 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

 
 

Figure 6 Variations of CR in 8 × 8 matrix 

 

As inferred from the plotted diagrams in the figures above, the consistency ratio (CR) 

will remain 0, enhancing the number of unknown values for matrices with perfect 

consistency. Furthermore, the blue line shows that for matrices with inconsistency, by 

enhancing the number of unknown elements, the consistency ratio (CR) decreases. The 

relation  (𝐶𝑅)ʹ <  CR holds through all points of the diagrams, where CR is the initial 

consistency of the matrix and (𝐶𝑅)ʹis the rate of consistency in the incomplete matrix. 
  

The obtained results are in accordance with Zhou et al. (2018). Moreover, the model 

proposed here could solve matrices with less than 30% unknown elements. 

 

 

5. Conclusion 

Considering that the core of the Analytic Hierarchy Process (AHP) is the availability of 

an information matrix for the decision-making process, an incomplete matrix resulting 

from a lack of information disrupts this process. Therefore, in this study, by concerning 

the incomplete information in the AHP and defining the variables and their relationship 

with each other, the mathematical model is investigated, and these relationships are 

provided in the form of a Multi-Objective Mathematical Programming Model. This study 

aims to present a model to estimate unknown matrix elements and the resulting weight 

priority vector by using goal programming and the similarity function. The developed 

model in this investigation can estimate 30% of the unknown elements. In Zhou et al.'s 

(2018) study, only the unknown elements are estimated. However, unknown elements 

and PCM weights were simultaneously calculated in this study. Moreover, the total 

weight priority vector was easily obtained.  

 

This study can be used in MCDM problems when there is the possibility of incomplete 

information from the experts. It can be beneficial in emergencies, where time is critical 

and access to complete information and application of decision-making procedures is 

time-consuming. Also, the state of uncertainty and interaction between factors are 

important to achieve better results in the MCDM technique. Therefore, we can make a 

compressive decision in future studies about these elements. 

 

0

0.02

0.04

0.06

0.08

1 2 3 4 5 6 7 8

C
R

 

Unknown Pairs 

Variation of CR 

Group 1

Group 2



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

18 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

 

APPENDIX 

Table A 

iPCM benchmark matrices used from Zhou et al. (2018)   

 
 Group 1: Group 2: 

4  4 

matrix 
[

1 0.8 1.55 1
1.25 1 ∗ ∗
0.65 ∗ 1 ∗
1 ∗ ∗ 1

] [

1 ∗ 4 ∗
∗ 1 2 ∗

0.25 0.5 1 2
∗ ∗ 0.5 1

] 

5  5 

matrix 

[
 
 
 
 

1 ∗ ∗ 5 8
∗ 1 3 5 ∗
∗ 0.33 1 ∗ 5

0.2 0.2 ∗ 1 3
0.13 ∗ 0.2 0.33 1]

 
 
 
 

 

[
 
 
 
 

1 2 2 4 8
0.5 1 ∗ 2 ∗
0.5 ∗ 1 2 ∗
0.25 0.5 0.5 1 ∗
013 ∗ ∗ ∗ 1]

 
 
 
 

 

6  6 

matrix 

[
 
 
 
 
 

1 5 ∗ 3 6 ∗
0.2 1 0.33 ∗ 3 ∗
∗ 3 1 0.5 ∗ 0.33

0.33 ∗ 2 1 5 ∗
0.17 0.33 ∗ 0.2 1 0.2
∗ ∗ 3 ∗ 5 1 ]

 
 
 
 
 

 

[
 
 
 
 
 

1 ∗ 2.25 ∗ 5.06 7.6
∗ 1 1.5 ∗ 3.38 5.06

0.44 0.67 1 ∗ 2.25 3.38
∗ ∗ ∗ 1 ∗ 2.25

0.2 0.3 0.44 ∗ 1 ∗
0.13 0.2 0.3 0.44 ∗ 1 ]

 
 
 
 
 

 

7  7 

matrix 

[
 
 
 
 
 
 

1 0.25 5 ∗ 0.33 0.5 ∗
4 1 ∗ 0.33 ∗ 0.25 ∗

0.2 ∗ 1 0.14 0.14 ∗ 0.33
∗ 3 7 1 ∗ 2 3
3 ∗ 7 ∗ 1 2 3
2 4 ∗ 0.5 0.5 1 ∗
∗ ∗ 3 0.33 0.33 ∗ 1 ]

 
 
 
 
 
 

 

[
 
 
 
 
 
 

1 ∗ ∗ ∗ 4 ∗ ∗
∗ 1 ∗ ∗ ∗ 8 8
∗ ∗ 1 2 4 8 8
∗ ∗ 0.5 1 2 4 4

0.25 ∗ 0.25 0.5 1 2 2
∗ 0.13 0.13 0.25 0.5 1 1
∗ 0.13 0.13 0.25 0.5 1 1]

 
 
 
 
 
 

 

8  8 

matrix 

[
 
 
 
 
 
 
 

1 ∗ ∗ 7 6 6 ∗ ∗
∗ 1 ∗ 5 ∗ 3 ∗ 0.14
∗ ∗ 1 4 3 3 0.17 ∗

0.14 0.2 0.25 1 1 ∗ 0.11 0.13
0.17 ∗ 0.33 1 1 1 ∗ 0.11
0.17 0.33 0.33 ∗ 1 1 0.11 0.17
∗ ∗ 6 9 ∗ 9 1 ∗
∗ 7 ∗ 8 9 6 ∗ 1 ]

 
 
 
 
 
 
 

 

[
 
 
 
 
 
 
 

1 2 0.5 2 ∗ 2 0.5 ∗
0.5 1 ∗ 1 0.25 1 ∗ ∗
2 ∗ 1 4 ∗ 4 1 ∗

0.5 1 0.25 1 0.25 ∗ ∗ 1
∗ 4 ∗ 4 1 4 1 4

0.5 1 0.25 ∗ 0.25 1 0.25 ∗
2 ∗ 1 ∗ 1 4 1 ∗
∗ ∗ ∗ 1 0.25 ∗ ∗ 1]

 
 
 
 
 
 
 

 

 

  



IJAHP Article: Bagheri Fard Sharabiani, Gholamian, Ghannadpour/A new solution for 

incomplete AHP model using goal programming and similarity function 

 International Journal of the 
Analytic Hierarchy Process 

19 Vol 15 Issue 1 2023 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v15i1.1003 

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