The analytic hierarchy process with personalized individual semantics

Qiuxiang Zhou; Yucheng Dong; Hengjie Zhang; Yuan Gao

doi:10.2991/ijcis.11.1.34

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Volume 11, Issue 1, 2018, Pages 451 - 468

The analytic hierarchy process with personalized individual semantics

Authors

Qiuxiang Zhou¹^,qxzhou@stu.scu.edu.cn, Yucheng Dong¹^,ycdong@scu.edu.cn, Hengjie Zhang²^{, *}^,hengjiezhang@hhu.edu.cn, Yuan Gao¹^,gaoyuan1984@scu.edu.cn

¹Business School, Sichuan University, Chengdu 610065, China

²Business School, Hohai University, Nanjing 211100, China

^* Corresponding author

Corresponding Author

Hengjie Zhanghengjiezhang@hhu.edu.cn

Received 25 July 2017, Accepted 22 December 2017, Available Online 1 January 2018.

DOI: 10.2991/ijcis.11.1.34 How to use a DOI?
Keywords: Analytic hierarchy process (AHP); personalized individual semantics (PIS); numerical scale; the 2-tuple linguistic model; consistency-driven methodology
Abstract: Personalized individual semantics (PIS) is not unusual in our daily life, and it has an important influence on the final decision results in linguistic decision making. The analytic hierarchy process (AHP) has now become a popular decision tool because of its sound mathematical design and ease of applicability to real-world decision making problems. In the AHP, two formats of preference information are included: linguistic and numerical preference information. In order to implement the computation operation, the linguistic preference information is often transformed into the numerical preference information using a fixed numerical scale function (e.g., the Saaty scale). However, the PIS is not taken into account by the AHP with a fixed numerical scale function. Therefore, this study proposes a novel AHP framework with the PIS, and develops a consistency-driven methodology to minimize the inconsistency level of numerical preference information that transformed from linguistic preference information. In the proposed AHP framework, a two-stage based optimization model is designed to deal with the PIS, and the proposed optimization models are converted into some linear programming models that can be easily solved. Finally, a practical example and a comparison experiment are proposed to verify the validity of our proposal.
Copyright: © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

The analytic hierarchy process (AHP) that initialized by Saaty (1980) provides decision makers with a way to transform subjective judgments into objective measures[38]. Due to its mathematical simplicity and flexibility, AHP has been a favorite decision tool for research in many fields [43], such as manufacture, politics and economy, power and energy, education, and medicine. In general, the implementation of the AHP includes the following steps: (1) construct a decision-making problem in a hierarchy as goal, criteria, sub-criteria, and decision alternatives; (2) provide linguistic preference information (i.e., linguistic pairwise comparison matrices (LPCMs), or linguistic preference relations) at each level of the hierarchy structure; (3) transform the linguistic preference information into numerical preference information (i.e., numerical pairwise comparison matrices (NPCMs)) using a numerical scale function (i.e., the Saaty scale); (4) check the consistency of numerical preference information; (5) obtain the local priority vectors corresponding to the criteria or sub-criteria applying the given prioritization method (e.g., Eigenvalue method); (6) obtain the global priority vectors of alternatives by synthesizing the local priority vectors. The detailed information of the AHP can be found in [13, 38, 40].

By analyzing the implementation process of the AHP, we find that the validity of which depends on the selection of numerical scale function in step 3. This issue has been investigated by several scholars, and several numerical scale functions have been proposed, such as the Saaty scale function [38], the Ma-Zheng scale function [30], the geometrical scale function [21, 26, 29], and the Salo Hämäläinen scale function [37]. However, there are still some disputes on which numerical scale function is better. Harker and Vargas [23] argued that the choice of numerical scale function in the AHP is an open problem.

The selection of numerical scale in the AHP is closely related to the computing with words (CWW). In the field of CWW, there is a fact that the same word usually means different things for different experts ([31, 32]). For example, two experts may all provide their comments “good” on a project in the project review process. However, the term “good” often refers to different scores for these two experts when they grade the project on a scale of 0 to 100, one expert might give 80 and another give 90. There are two mainstreams in the literature to deal with this issue: (1) the type-2 fuzzy sets [31], (2) the multi-granular linguistic models [16, 34, 45, 50]. Despite these approaches are quite useful, they do not present the specific semantics for each expert. To overcome this problem, Li et al. [27] proposed an optimization-based model to deal with personalized individual semantics (PIS for short) in the linguistic decision making.

Based on the above analysis, there is no doubt that the PIS will occur in the AHP, and it has a very important influence on the final decision results. To our knowledge, the PIS is rarely considered by the existing literature except two works [25, 35]. In [25, 35], the existence of PIS in the AHP has been demonstrated by using empirical study. However, an effective way to deal with the PIS in the AHP is not provided in [25, 35].

The main purpose of this study is to propose a novel AHP framework: the AHP framework with the PIS. In the novel AHP, the most important step is the design of the transformation method between the linguistic and numerical preference information. To support the novel framework, we devise a consistency-driven approach that minimizes the consistency index of the transformed numerical preference information. This approach is based on a basic idea that if linguistic preference information is of acceptable consistency, then the consistency level of the transformed numerical preference information should be as better as possible, due to the fact that both the linguistic and numerical preference information are associated with the same expert. Specifically, we propose a two-stage based optimization model to address the PIS in the AHP. The first-stage optimization model is used to generate the individual numerical scales by maximizing the consistency level of the transformed numerical preference information, and the second-stage optimization model is utilized to further optimize the optimal solutions derived from the first-stage model. In addition, both the optimization models in first and second stages are converted into linear programming models that can be easily solved.

The remainder of this study is arranged as follows. Section 2 introduces the 2-tuple linguistic model, numerical scales and the AHP with a fixed numerical scale function. Section 3 designs a novel AHP framework with the PIS. Next, Section 4 presents the consistency-driven approach to deal with the PIS in the AHP. Following this, a practical example is used to illustrate the application process of our proposal. Section 6 gives a comparative study. Finally, Section 7 concludes the study and discusses the future research directions.

2. Preliminaries

In this section, the fundamental knowledge regarding the 2-tuple linguistic model, preference relations, numerical scales, and the AHP with a fixed numerical scale, are briefly described.

2.1. The 2-tuple linguistic model and preference relations

(1) The 2-tuple linguistic model

Let S= {s_i|i = 0,1,2,...,g} be a linguistic term set, in which the number of linguistic terms is odd (i.e., g+1). The term s_i denotes a possible value for a linguistic variable, and the following two conditions should be satisfied [15].

(i)
If i > j, then s_i > s_j.
(ii)
We have Neg (s_i) = s_j such that j = g – i, where Neg denotes a negation operator.

The 2-tuple linguistic model that proposed by Herrera and Martínez [24] is a popular model for computing with words. Let S be as above. In the 2-tuple linguistic model, a 2-tuple (s_i, α) is utilized to express the equivalent information to β ∈ [0, g]: Δ :[0, g] → S × [−0.5, 0.5), where

Δ(β)=(si,α),with{si,i=round(β)α=β−i,α∈[−0.5,0.5)

where Δ is a one to one mapping function. Let S¯={(si, α)|si∈S,α∈[−0.5,0.5)} be the 2-tuple linguistic term set. In addition, Δ has an inverse function Δ−1:S¯→[0,g] with Δ⁻¹(s_i,α) = i + α.

(2) Preference relations

Let X = {x₁, x₂,...,x_n} be a set of alternatives. During the decision making process, preference relations (e.g. [7, 28, 22]) are very useful tools for decision makers to express their preferences over X. In particular, linguistic and multiplicative preference relations are widely adopted in the decision making. In the following, we introduce these two types of preference relations.

Definition 1 [1, 12]:

The linguistic preference relation is represented by a matrix L = (l_ij)_n×n, in which the element l_ij ∈ S signifies the preference intensity of alternative x_i over x_j. The linguistic preference relation L = (l_ij)_n×n is often assumed to be reciprocal, that is l_ij = Neg(l_ji) for i, j = 1,...,n.

Definition 2 [40, 46]:

The multiplicative preference relation is denoted using a matrix A = (a_ij)_n×n, in which a_ij > 0 signifies the important degree (or preference intensity) of alternative x_i over x_j and a_ij × a_ji = 1 for i, j = 1,….,n. If a_it × a_tj = a_ij (∀ i, j, t = 1,2,...,n), then A is considered as a completely consistent multiplicative preference relation.

2.2. Numerical scales

The numerical scale was developed by Dong et al. [19], which is utilized to convert linguistic information into numerical information.

Definition 3 [19]:

Let S be as before, and R be the real number set. The numerical scale of S is defined as a function: NS : S → R, and we use NS(s_i) to denote the numerical index of linguistic term is s_i.

Moreover, the numerical scale of the 2-tuple linguistic information was also developed, which is defined as follows.

Definition 4 [19]:

Let S, S¯ and NS be as above. The numerical scale function NS¯ on S¯ for (si,α)∈S¯ , is defined as below:

NS¯((si,α))={NS(si) + α NS(si+1)−NS(si)), α≥0 NS(si) + α NS(si)−NS(si−1)), α<0

For simplification, NS¯ will also be denoted as NS throughout this study.

2.3. Analytic hierarchy process

This section introduces the basic framework of AHP, and the 2-tuple linguistic modeling for the AHP scale problem.

(1) The framework of AHP

Saaty [38] developed following steps for applying the AHP.

a) Establish the hierarchical structure

When using AHP to solve a decision problem, firstly, decision makers need to disassemble the problem into a hierarchy of more easily comprehended sub-problems. A hierarchical structure (shown as Fig. 1) is often composed by the following levels: goal level, criterion levels, and alternative level. In addition, there are subordinate relationships between the elements on two adjacent levels in the hierarchy.

Let H_k (k = 1,...,m) be the k th level of the given hierarchical structure. The top level (i.e., H₁) is the goal level, the bottom level (i.e., H_m) is the alternative level, the others are criterion levels. Let n_k (k = 1,...,m) be the number of elements (criteria/alternatives) in H_k. In particular, n_m denotes the number of alternatives. Let ekz(z=1,…,nk,k=1,…,m) be the z th element in H_k; nkz(z=1,...,nk,k=1,...,m−1) be the number of the elements in H_k+1 controlled by the element ekz ; and eik,z(i=1,...,nkz, z=1,...,nk, k=1,...,m−1) be the i th element controlled by the element ekz .

b) Make pairwise comparisons using AHP linguistic scale to obtain linguistic preference relations

Decision makers provide their linguistic preference information regarding criteria and alternatives corresponding to the parent element at the adjacent upper level. For the parent element ekz(z=1,...,nk,k=1,...,m−1) , decision makers use AHP linguistic term set to characterize the important degree between the element eik,z and ejk,z (i,j=1,...,nkz) , and then construct the LPCM Lkz=(lijk,z)nzk×nzk (z=1,…,nk,k=1,…,m−1) .

c) Transform LPCMs into NPCMs using AHP numerical scale

In the traditional AHP, decision makers employ a fixed numerical scale (i.e., the Saaty scale) to transform the LPCMs Lkz into NPCMs Akz=(aijk,z)n×n(z=1,…,nk, k=1,…, m−1) .

d) Employ a prioritization method to derive the local priority vectors of criteria and alternatives

By adopting a selected prioritization method, we can derive priority vectors lpkz=(lp1k,z,…,lpnkzk,z)T (z=1,…,nk, k=1,…,m−1) from the NPCMs Akz . lpik,z(i=1,…,nkz) represents the local priority of the element eik,z with respect to the element ekz , all of elements eik,z(i=1,…,nkz) are in H_k+1 and controlled by the element ekz . For the elements in H_k+1 that aren’t controlled by the element ekz , the local priorities are 0. Then, we can obtain the local priority vector of all elements in H_k+1 with respect to the element ekz , defined as Pkz=(p1k,z,…,pnk+1k,z)T(z=1,…,nk,k=1,…,m−1) , pjk,z(j=1,…,nk+1) represents the local priority of the element ek+1j with respect to the element ekz .

To derive a priority vector from a NPCM A = (a_ij)_{n× n}, the prioritization method is often used. There are several prioritization methods (e.g., [8, 9, 33, 38, 39, 41, 49]). In this paper, we adopt the Eigenvalue method proposed by Saaty [39], which is introduced below.

Let λ and p′ be the eigenvalue and eigenvector of matrix A, respectively, and they can be obtained by the following model:

(1)Ap′=λp′,eTp′=1,

Saaty sets the principle eigenvector as the desired priority vector of A.

e) Consistency test

Consistency test is adopted to check whether the NPCMs are logical or not. If the NPCMs fail the consistency test, decision maker should adjust the corresponding LPCMs. There are many consistency improving approaches [5, 47]. In particular, the consistency index (CI) based on the eigenvalue prioritization method proposed by Saaty [38] is calculated as below:

(2)CI=λmax−nn−1

where λ_max and n denote the maximum eigenvalue and the size of NPCM A, respectively.

Further, Saaty proposed the concept of consistency ratio, that is

(3)CR=CIRI

where RI denotes the average value of CI obtained from a group of randomly generated NPCMs. In general, if CR ≤ 0.1, then the NPCM is considered as consistent. The RI values for different NPCM sizes (n) are given in Table 1.

n	3	4	5	6	7	8	9
RI	0.52	0.89	1.12	1.26	1.36	1.41	1.46

Table 1.

Values of RI for different matrix size (n)

In addition, there are several other approaches for measuring the consistency level of NPCMs (see [2, 3, 6, 14, 42]).

f) Synthesize the priority vectors to yield the global priority vector

This process is to synthesize the local priority vectors with respect to all criteria to generate the global priority vectors. Further, we can obtain the ranking of the alternatives from the global priority vector at the alternative level.

Suppose that wk−1=(w1k−1,w2k−1,…,wnk−1k−1)T(k=2,3,…,m) is the global priority vector of the elements in H_k−1 with respect to the global goal. Laying out the local priority vectors of all elements in H_k corresponding to the each element in H_k−1 into a matrix, defined as Pk=(Pk−11,Pk−12,…,Pk−1nk−1)=(pijk)nk×nk−1 (k=2,…,m) , then the global priority vector of elements in H_k can be obtained by,

(4)wk=(w 1k,w 2k,…,w nk−1k)T=Pkwk−1

or wik=∑j=1k−1pijk×wjk−1,i=1,…,n;k=2,3,...,m

Therefore, w^k = P^kP^k−1 … w² and w2=P2=P11=(p11,1,p21,1,…,pn21,1)T . The ranking of alternatives can be obtained according to w^m.

(2) 2-tuple linguistic modeling of the AHP scale problem

The scale in AHP is composed of AHP linguistic and numerical scales. Table 2 shows the Saaty’s linguistic scale (see [40]).

Grade	AHP linguistic scale
1	Equally important
2	Weakly more important
3	Moderately more important
4	Moderately plus more important
5	Strongly more important
6	Strongly more important
7	Demonstratedly more important
8	Very, very Strongly more important
9	Extremely more important

Table 2.

Linguistic part of the Saaty scale

According to Saaty’s linguistic scale, we define the following linguistic term set [12]:

S^AHP = {s₀ = extremely less important, s₁ = very very strongly less important, s₂ = demonstratedly less important, s₃ = strongly plus less important, s₄ = strongly less important, s₅ = moderately plus less important, s₆ = moderately less important, s₇ = weakly less important, s₈ = equally important, s₉ = weakly more important, s₁₀ = moderately more important, s₁₁ = moderately plus more important, s₁₂ = strongly more important, s₁₃ = strongly plus more important, s₁₄ = demonstratedly more important, s₁₅ = very very strongly more important, s₁₆ = extremely more important}

where s_i (i = 0,1,2,...,16) satisfies the characteristics of linguistic term set, and {s_i, s_16-i} (i = 8,9,...,16) is associated with the (i − 7) th gradation of the Saaty’s linguistic scale.

There are 17 numerical values in the AHP numerical scale, which are described below:

(5){1fi, f1,fi}, i=2,3,…,9,

where f₁ =1 and f_i+1 > f_i > 1. The value of f_i (i = 1, 2,...,9) is associated with the i th gradation of the AHP linguistic scale. Naturally, there are different numerical scales when setting different values for f_i (i = 2, 3,...,9) (e.g., [21, 23, 26, 30, 37]).

In [13], based on the 2-tuple linguistic model, the AHP numerical scale is developed, which is defined as below:

Definition 5 [13]:

Let SAHP¯ be the 2-tuple linguistic term set corresponding to S^AHP (i.e., SAHP¯={(s,α)|s∈SAHP,α∈[−0.5,0.5)} ). The function NSAHP:SAHP¯→R+ is called an AHP numerical scale function, which satisfies: (1) NS^AHP (s) × NS^AHP (Neg(s)) = 1, for any s∈SAHP¯ , and (2) NS^AHP (s_i) < NS^AHP (s_j) if i < j.

Based on the AHP numerical scale function, we can transform an LPCM L = (l_ij)_n×n into a NPCM A = (a_ij)_n×n, where a_ij = NS^AHP (l_ij) for i, j = 1, 2, …..,n.

3. The framework of the AHP with PIS

By taking into account an un-negligible factor, PIS, in the AHP, this section proposes a novel AHP framework: the AHP framework with PIS.

As introduced in Section 2, in the traditional AHP, a decision maker provides LPCMs in the established hierarchical structure. Then, for conducting computation operation, LPCMs are transformed into NPCMs using a certain numerical scale function. The choice of numerical scale has an important influence on the final decision results, and there are several common numerical scale functions (e.g., [21, 26, 29, 30, 37, 38]). When selecting a numerical scale function to quantify the LPCMs (i.e., the transformation between linguistic information and numerical information), the PIS is an un-negligible factor and it has an important influence on the final decision results. Huizingh and Vrolijk [25], and Poyhonen et al. [35] have demonstrated the existence of PIS in the AHP. Recently, Li et al. [27] researched the PIS in linguistic decision making. Motivated by the challenge of modelling the PIS in AHP, and inspired by the works of Saaty [38], Huizingh and Vrolijk [25], Poyhonen [35], and Li et al. [27], we propose a novel AHP framework: AHP framework with the PIS. The proposed framework is described in Fig. 2.

There are two differences between AHP with the PIS and the AHP with the fixed numerical scale (FNS): (1) consistency test. In the traditional AHP with a fixed numerical scale, the consistency test is conducted on NPCMs, and in the AHP with the PIS, it is to check the consistency of LPCMs; (2) the transformation process between LPCMs and NPCMs. In the AHP with the FNS, the transformation process is based on a certain numerical scale (e.g., the Saaty scale). In the AHP with the PIS, the transformation between LPCMs and NPCMs is conducted by the individual numerical scales.

(1) Consistency test regarding LPCMs

Up to now, a lot of works regarding the consistency issue of LPCMs have been reported (see [1], [44], [47] and [48]). In particular, transitive properties have been utilized to capture the consistency issues of LPCMs. Let L = (l_ij)_n×n be as above. In the following, we introduce several transitive properties of L = (l_ij)_n×n (see [18]):

a) Intransitivity. l_ij ≥ s_g/2, l_jt ≥ s_g/2 ⇒ l_it < s_g/2 for ∀i, j, t.

Example 1.

Let L = (l_ij)_3×3 be an LPCM over alternatives {x₁, x₂, x₃}, and L is listed below.

L=(s8s10s5s6s8s13s11s3s8 )

In an LPCM L, x₁ is moderately more important than x₂, i.e., l₁₂ = s₁₀ > s₈, and x₂ is strongly plus more important than x₃, i.e., l₂₃ = s₁₃ > s₈, and x₁ is moderately plus more important than x₃, i.e.,l₁₃ = s₅ > s₈. Thus, L is an LPCM with intransitivity.

b) Weak stochastic transitivity. l_ij ≥ s_g/2, l_jt ≥ s_g/2, ⇒ l_it ≥ s_g/2 for ∀i, j, t.

Example 2.

Let L = (l_ij)_3×3 be an LPCM over alternatives {x₁, x₂, x₃}, and L is listed below.

L=(s8s10s11s6s8s13s5s3s8 )

In an LPCM L, x₁ is moderately more important than x₂, i.e., l₁₂ = s₁₀ > s₈, x₂ is strongly plus more important than x₃, l₂₃ = s₁₃ > s₈, and x₁ is moderately plus more important than x₃, l₁₃ = s₁₁ > s₈. Thus, L is an LPCM with intransitivity. Weak stochastic transitivity.

c) Strong stochastic transitivity. l_ij ≥ s_g/2, l_jt ≥ s_g/2 ⇒ l_it ≥ max (l_ij, l_jt) for ∀i, j, t.

Example 3.

Let L= (l_ij)_3×3 be an LPCM over alternatives {x₁, x₂, x₃}, and L is listed below.

L=(s8s10s14s6s8s13s11s3s8 )

In an LPCM L, x₁, is moderately more important than x₂, i.e., l₁₂ = s₁₀ > s₈, x₂, is strongly plus more important than x₃, i.e., l₂₃ = s₁₃ > s₈, and x₁ is demonstrately more important than x₃, i.e., l₁₃ = s₁₄ > max{s₁₀, s₁₃}. Thus, L is an LPCM with strong stochastic transitivity.

d) Additive transitivity. Δ−1(lij)+Δ−1(ljt)−g2=Δ−1(lit) for ∀i,j,t.

Example 4.

Let L = (l_ij)_3×3 be an LPCM over alternatives {x₁, x₂, x₃}, and L is listed below.

L=(s8s10s15s6s8s13s11s3s8 )

In an LPCM L, x₁ is moderately more important than x₂, i.e., l₁₂ = s₁₀ > s₈, x₂ is strongly plus more important than x₃, i.e., l₂₃ = s₁₃ > s₈, and x₁ very very strongly important than x₃ i.e., l₁₃ = s₁₅ = Δ⁻¹ (s₁₀) + Δ⁻¹(s₁₃) – s₈. Thus, L is an LPCM with additive transitivity.

Obviously, the condition of the additive transitivity is the most restrictive, and followed by the strong stochastic transitivity and the weak stochastic transitivity. Usually, if a set of pre-established transitive properties are satisfied, we then consider that the consistency level of an LPCM is acceptable. In this paper, we assume that the consistency level of an LPCM is acceptable when the weak stochastic transitivity is satisfied. To improve the consistency level of an LPCM, several approaches have been suggested (e.g., [4, 5, 47]).

(2) Consistency-driven approach to obtain the numerical scale with PIS

In the AHP, the LPCMs given by a decision maker are often transformed into NPCMs for implementing the computation operation. Clearly, both the LPCMs and NPCMs are the preferences associated with the same decision maker. Naturally, if the consistency level of LPCMs is acceptable, then the consistency level of the transformed NPCMs should be as better as possible. Following this idea, we put forward a consistency-driven approach to set numerical scale with PIS. In the consistency-driven approach, a two-stage based optimization model is established to generate the individual numerical scales. The first-stage optimization model is to minimize the overall inconsistency level of NPCMs that transformed from LPCMs, and the second-stage optimization model is to obtain the unique optimal solution by further optimizing the optimal solution(s) obtained from the first-stage model.

The consistency-driven approach is formally presented in Section 4.

Discussion. Clearly, the consistency-driven approach is a two-procedure based approach. In the work of Dong et al. [13], a two-procedure based approach is also reported to set numerical scales in the AHP. In the following, the differences between our proposal and the work of Dong et al. [13] are analyzed:

(1)
The first procedure is the detection of the individual characteristics in Dong et al. [13], and the first procedure of our proposal is the consistency test of LPCMs.
(2)
In the second procedure of Dong et al. [13], a nonlinear programming model is constructed to yield individual numerical scales by optimally matching the individual characteristics generated from the first procedure. In the second procedure of our proposal, a consistency-driven optimization model that minimizes the inconsistency index of the transformed NPCMs is built to produce the individual numerical scales. In addition, the consistency-driven optimization model is converted into a linear programming model that can be easily solved.

4. Consistency-driven methodology to deal with the PIS

This section presents a consistency-driven approach to address the PIS in AHP. Specifically, we propose a consistency-driven optimization model to yield the individual numerical scales in section 4.1. Moreover, we make some discussions regarding the optimal solution(s) derived from the consistency-driven optimization model.

4.1. Consistency-driven optimization model

Recall that Lkz=(lijk,z)nkz×nkz (i,j=1,…,nkz, z=1,…,nk, k=1,…,m−1) are the LPCMs in the established hierarchical structure. In the consistency-driven methodology, the LPCMs Lkz=(lijk,z)nkz×nkz , (z = 1, …, n_k, k = 1, …, m−1) are assumed to be consistent. There are several approaches (e.g., [4, 5]) to improve the consistency level of the LPCM. Let Akz=(aijk,z)nkz×nkz be the NPCM associated with LPCM Lkz=(lijk,z)nkz×nkz . We know that both Lkz and Akz(z=1,…,nk,k=1,…,m−1) are the preferences associated with the same decision maker. Naturally, we hope that the consistency level of Akz should be as better as possible, due to the fact that the LPCM Lkz(z=1,…,nk,k=1,…,m−1) is of acceptable consistency.

Before proposing the consistency-driven optimization model, the consistency index (CI) of a NPCM is introduced.

For a NPCM A = (a_ij)_n×n, if a_it ×a_tj = a_ij for ∀ i,j,t ∈ {1,2,…,n}, then A is a completely consistent NPCM. In this case, log(a_it)+log(a_tj) = log(a_tj), ∀ i, j, t ∈ {1,2,…,n}. Further, we have ∑i,j,t=1,i<t<jn|log(ait)+log(atj)−log(atj)|=0 , if A is a completely consistent NPCM. However, in practice, it is difficult to provide a completely consistent NPCM for a decision maker. So, consistency measures are often used. In the following, the consistency level of a NPCM is defined.

Definition 6:

The consistency level of NPCM A = (a_ij)_n×n can be defined as follows:

(6)CL(A)=1n(n−1)(n−2)∑i,j,t=1,i<t<jn|log(ait)+log(atj)−log(aij)|.

Clearly, CL(A) ∈ [0,1]. The smaller the CL(A) value, the better the consistency level of A is.

Naturally, we hope that the transformed NPCMs Akz(z=1,…,nk,k=1,…,m−1) are as consistent as possible, i.e.,

(7)min∑k=1m−1∑z=1nkCI(Akz).

Meanwhile, the AHP numerical scale NS^AHP should satisfy the following two conditions (described as (8) and (9)):

(8)NSAHP(si)≤NSAHP(si+1), i=9,10,…,16,

and

(9)NSAHP(si)×NSAHP(s16−i)=1, i=9,10,…,16.

If we set NSAHP(si)={19−i,i=0,1,2,…7i−7,i=8,9,…16 , then it is the Saaty scale. To make a linkage with the Saaty scale, we adopt the following way to set the range of the individual numerical scales:

(10)i−7−Δ≤NSAHP(si)≤i−7+Δ, i=9,10,…,16.

where, Δ ∈ [0,1] is a parameter. Clearly, if we set Δ = 0, then NS^AHP is the Saaty scale.

Take logarithms on both sides, (8)–(10) can be equivalently written as (11)–(13), respectively:

(11)log(NSAHP(si))≤log(NSAHP(si+1)), i=9,10,…,16

(12)log(NSAHP(si))+log(NSAHP(s16−i))=0, i=9,10,…,16

(13)log(i−7−Δ)≤log(NSAHP(si))≤log(i−7+Δ), i=9,10,…,16

According to (7) and (11)–(13), we propose a consistency-driven optimization model:

(14)min∑k=1m−1∑z=1nkCI(Akz)=∑k=1m−1∑z=1nk1nkz(nkz−1)(nkz−2)∑i,t,j=1,i<t<jnkz|log(aitk,z)+log(atjk,z)−log(aijk,z)|s.t. {aijk,z=NSAHP(lijk,z), i,j=1,…nkz, z=1,…nk,k=1,…m−1log(NSAHP(si))≤log(NSAHP(si+1)), i=9,10,…,16log(NSAHP(si))+log(NSAHP(s16−1))=0, i=9,10,…,16log(i−7−Δ)≤log(NSAHP(si))≤log(i−7+Δ), i=9,10,…,16

In model (14), NS^AHP (s_i) (i = 0,1,…, 16) are decision variables. Solving model (14) yields the AHP numerical scale NS^AHP(s_i) (i = 0,1,…, 16). Based on the obtained NS^AHP(s_i) (i = 0,1,…, 16), LPCMs can be transformed into NPCMs.

For convenience, let p(s) (s ∈ S^AHP) be the position index of s. For instance, we have p(s_i) = i. Let Bⁱ = log(NS^AHP(s_i)), then model (14) can be written as the following model:

(15)min∑k=1m−1∑z=1nk1nkz(nkz−1)(nkz−2)∑i,t,j=1,i<t<jnkz|Bp(litk,z)+Bp(ltjk,z)−Bp(lijk,z)|s.t. {Bp(lijk,z)=log(NSAHP(sp(lijk,z))), i,j=1,…nkz, z=1,…,nk,k=1,…,m−1Bi−Bi+1≤0, i=9,10,…,16Bi−B16−i=0, i=9,10,…,16log(i−7−Δ)≤Bi≤log(i−7+Δ), i=9,10,…,16

where Bⁱ (i = 0,1, 2,...,16) are decision variables.

In the following, we propose a method to transform model (15) into a linear programming model, which is described as Theorem 1.

For simplicity, we set K = {1, …, m-1}, N^k = {1,...,n_k} (k ∈ K), and Qkz={1,...,nkz×(nkz−1)×(nkz−2)/6} , z ∈ N_k, k ∈ K.

Theorem 1.

Let MqkZ be a variable, where |log(aitk,z)+log(atjk,z)−log(aijk,z)|≤MqKZ(qkz∈Qkz,z∈Nk,k∈K) . Model (15) can be equivalently transformed into the following linear programming model.

(16)min∑k=1m−1∑z=1nk1nkz(nkz−1)(nkz−2)∑qkz=1nkz×(nkz−1)×(nkz−2)/6Mqkzs.t. {Bp(litk,z)+Bp(ltjk,z)−Bp(lijk,z)≤Mqkz, (i,j,t=1,…,nkz,i<t<j,qkz∈Qkz,z∈Nk,k∈K) 16−1 −Bp(litk,z)−Bp(ltjk,z)−Bp(lijk,z)≤Mqkz, (i,j,t=1,…,nkz,i<t<j,qkz∈Qkz, z∈Nk, k∈K) 16−2Mqkz≥0, (qkz∈Qkz, z∈Nk, k∈K) 16−3Bp(lijk,z)=log(NSAHP(sp(lijk,z))), i,j=1,…,nkz 16−4Bi−Bi+1≤0, i=9,10,…,16 16−5Bi−B16−i≤0, i=9,10,…,16 16−6log(i−7−0.4)≤Bi≤log(i−7+0.4), i=9,10,…,16 16−7

Proof.

In model (16), constrains (16-1), (16-2), and (16-3) guarantee that Mqkz≥ |log(aitk,z)+log(atjk,z)−log(aijk,z)| . Moreover, any feasible solution that satisfies Mqkz> |log(aitk,z)+log(atjk,z)−log(aijk,z)| is not the optimal solution, based on the objective function of model (16). So, constrains (16-1), (16-2), and (16-3) ensure that Mqkz= |log(aitk,z)+log(atjk,z)−log(aijk,z)| . Therefore, model (15) can be transformed into (16). This completes the proof of Theorem 1.

4.2. Further discussion

This section discusses the problem of uniqueness of optimal solution to model (14).

In Section 4.1, we can obtain the optimal solution(s) to model (14). Nevertheless, in some cases, the optimal solution(s) to model (14) may not be unique. In particular, some of them are unreasonable (in the sense of consistency). To demonstrate this issue, Example 5 is provided.

Example 5.

Suppose that there is a hierarchical structure (see Fig. 3), which consists of one goal {G}, three criteria {C₁, C₂, C₃}, and three alternatives X = {x₁, x₂, x₃}. The LPCMs given by a decision maker based on S^AHP are listed as follows,

L11=(s8s4s10s12s8s14s6s2s8 )L21=(s8s7s8s9s8s10s8s6s8 )L21=(s8s7s16s9s8s16s0s0s8 )L22=(s8s9s6s7s8s1s10s15s8 )

In this example, we set Δ = 0.4. Using model (14), we can obtain a set of optional solutions. Here, we only list two optimal solutions: Ω₁: {NS^AHP(1) (s_i), i = 0,1,…, 16} and Ω₂: {NS^AHP(2) (s_i), i = 0,1,…, 16}, and they are provided in Table 3.

	8	9	10	11	12	13	14	15	16
Ω₁:NS^AHP(s_i)	1	2.29	2.29	3.57	4.38	5	7.28	7.28	9.19
Ω₂:NS^AHP(s_i)	1	2	2	3.60	4.51	5	7.03	7.03	9.6

Table 3.

The individual numerical scales in Ω₁ and Ω₂

Based on Ω₁ and Ω₂, L11 , L21 , L22 , and L23 can be transformed into NPCMs A11 , A21 , A22 , and A23 , and they are listed below:

(1)
the NPCMs Akz(1)(k=1,2,z=1,…,nk) under Ω₁
A11(1)=(10.232.294.3817.280.440.141 ),A21(1)=(10.4412.2912.2910.441 ),A22(1)=(12.290.440.4410.142.297.281 ),A23(1)=(10.449.192.2919.190.110.111 ).
(2)
the NPCMs Akz(2)(z=1,…,nk,k=1,2) under Ω₂
A11(2)=(10.2224.5117.030.50.141 ),A21(2)=(10.5121210.51 ),A22(1)=(120.50.510.1427.031 ),A23(2)=(10.59.62.19.60.10.11 ).

According to definition 6, we can obtain the consistency levels of the transformed NPCMs Akz(1) and Akz(2)(z=1,…,nk,k=1,2) under Ω₁ and Ω₂, respectively, which are listed in Table 4.

	CI(A21)	CI(A21)	CI(A22)	CI(A23)
Ω₁	0.08	0	0.05	0.14
Ω₂	0.06	0	0.09	0.12

Table 4.

The consistency levels of NPCMs under Ω₁ and Ω₂, respectively

Example 5 shows that: (1) the optimal solutions to model (14) are not unique; and (2) maxk,z{CI(Akz(1))}≥maxk,z{CI(Akz(2))} , this implies that the optimal solution Ω₂ is better than Ω₁, in the sense of consistency. Therefore, to further optimize the optimal solutions to model (14) is necessary.

Let M^* be the optimal objective value to model (14), let Ω = {Ω₁, Ω₂,…,Ω_n} (Ω_r = {NS^AHP(r)(s₀), …, NS^AHP(r)(s₁₆)}, r = 1,2,…,n) be a set of optimal solutions to model (14) let Akz(r)(z=1,2,...,nk,k=1,2,...m−1) be the transformed NPCMs under the optimal solution Ω_r (r = 1,2,…, n). Then, using the optimization model

(17)minΩr ∈Ω maxk∈K,z∈Nk{ CI(A kz(r)) }

generates the unique optimal solution. Model (17) further minimizes the maximal consistency index of the transformed NPCMs.

Further, model (17) can be rewritten as follows:

(18)min maxk∈K, z∈Nk{CI(Akz)}=minmaxk∈K,z∈Nk{1nkz(nkz−1)(nkz−2)∑i,j,t=1,i<t<jnzk|log(aitk,z)+log(atjk,z)−log(aijk,z)|}s.t.{aijk,z=NSAHP(lijk,z), i,j=1,…,nkz,z=1,…,nk,k=1,…,m−1 18−1∑k=1m−1∑z=1nk1nkz(nkz−1)(nkz−2)∑i,j,t=1,i<t<jnzk|log(aitk,z)+log(atjk,z)−log(aijk,z)|=M* 18−2log(NSAHP(si))≤log(NSAHP(si+1)), i=9,10,…,16 18−3log(NSAHP(si))+log(NSAHP(s16−i))=0, i=9,10,…,16 18−4log(i−7−Δ)≤log(NSAHP(si))≤log(i−7+Δ), i=9,10,…,16 18−5

In model (18), NS^AHP (s_i) (i = 0,…, 16) are decision variables. The constraints (18-1)–(18-5) guarantee that the optimal solutions of model (18) are subsets of the feasible solution of model (14), constraints (18-3), (18-4) guarantee that the AHP numerical scale NS^AHP (s_i) (i = 0,…, 16) is ordered and reciprocal, constraint (18-5) is used to control the range of individual numerical scales.

Solving model (18) obtains the optimal individual numerical scales. Let p(s) (s ∈ S^AHP) and Bⁱ (i = 0,1, …,16) be as above. Model (18) can be equally written as model (19).

(19)minmaxk∈K,z∈Nk{CI(Akz)}s.t.{∑k=1m−1∑z=1nkCI(Akz)=M*CI(Akz)=1nkz(nkz−1)(nkz−2)∑i,j,t=1,i<t<jnzk|Bp(litk,z)+Bp(ltjk,z)−Bp(lijk,z)|,(i,j,t=1,…,nkz,i<t<j, z∈Nk, k∈K)Bp(lijk,z)=log(NSAHP(sp(lijk,z))), i,j=1,…nkz,z=1,…,nk,k=1,…,m−1Bi−Bi+1≤0, i=9,10,…,16Bi+B16−i=0, i=9,10,…,16log(i−7−Δ)≤Bi≤log(i−7+Δ), i=9,10,…,16

In model (19), Bⁱ (i = 0,1,2,…,16) are the decision variables. Solving model (19) obtains the optimal solution B^i,* (i = 0,1,2,…,16) to Bⁱ (i = 0,1,2,…,16). Further, we obtain the individual numerical scales by NS^AHP (s_i) = e^{B^i*} (i = 0,1, 2,...,16).

Model (19) is a non-linear programming model. To solve model (19) easily, Theorem 2 is proposed to transform model (19) into a linear programming model.

Theorem 2.

Let Mqkz and λ be two variables, and they satisfy: |log(aitk,z)+log(atjk,z)−log(aijk,z)|≤Mqkz(qkz∈Qkz,z∈Nk,k∈K) . Then, model (19) can be equally converted into the linear programming model (20).

(20)minλs.t.{CI(Akz)≤λ,(z∈Nk,k∈K) 20−1∑k=1m−1∑z=1nkCI(Akz)=M* 20−2CI(Akz)=1nkz(nkz−1)(nkz−2)∑qkz=1nkz×(nkz−1)×(nkz−2)/6Mqkz,(z∈Nk,k∈K) 20−3Bp(litk,z)+Bp(ltjk,z)−Bp(lijk,z)≤Mqkz, (i,j,t=1,…,nkz,i<t<j,qkz∈Qkz, z∈Nk,k∈K) 20−4 −Bp(litk,z)−Bp(ltjk,z)−Bp(lijk,z)≤Mqkz, (i,j,t=1,…,nkz,i<t<j,qkz∈Qkz, z∈Nk,k∈K) 20−5 Mqkz≥0,(qkz∈Qkz,z∈Nk,k∈K) 20−6Bp(lijk,z)=log(NSAHP(sp(lijk,z))), (i,j=1,…,nkz,z∈Nk,k∈K) 20−7Bi−Bi+1≤0, i=9,10,…,16 20−8Bi+B16−i=0, i=9,10,…,16 20−9log(i−7−0.4)≤Bi≤log(i−7+0.4), i=9,10,…,16 20−10

Proof.

Constraint (20-1) guarantees that λ≥maxk∈K,z∈Nk{CI(Akz)} . Moreover, any feasible solutions with λ>maxk∈K,z∈Nk{CI(Akz)} are not the optimal solutions to model (20), according to its objective function. As a result, we have λ=maxk∈K,z∈Nk{CI(Akz)} . Constraints (20-4), (20-5), (20-6) and the objective function in model (20) guarantee that |log(aitk,z)+log(atjk,z)−log(aijk,z)|=Mqkz . Thus model (19) can be equally converted into model (20).

5. Practical example

In this section, a practical example that presented in Srdjevic [41] is used to show the application process of our proposal.

This example focuses on the issue of reservoir water allocated to multiple purposes. The hierarchical structure that described as Fig. 4 is used by a decision maker to select profitable purpose. The hierarchical structure is made up of three levels: the goal level, the criterion level and the alternative level. The LPCMs expressed by the decision maker are provided in Tables 5–10.

Goal	National income	Regional gains	Balance of payment	Import substitution	Foreign exchange
National income	s₈	s₉	s₁₂	s₁₀	s₉
Foreign exchange	s₇	s₈	s₁₄	s₁₀	s₁₀
Balance of payment	s₄	s₂	s₈	s₅	s₄
Import substitution	s₆	s₆	s₁₁	s₈	s₁₀
Regional gains	s₇	s₆	s₁₂	s₆	s₈

Table 5.

LPCM with respect to reservoir storage allocation L11

National income	Electric power generation	Irrigation	Flood protection	Water supply	Tourism recreation	River traffic
Electric power generation	s₈	s₁₂	s₁₀	s₁₃	s₁₄	s₁₂
Irrigation	s₄	s₈	s₂	s₇	s₉	s₉
Flood protection	s₆	s₁₄	s₈	s₁₄	s₁₀	s₁₁
Water supply	s₃	s₉	s₂	s₈	s₇	s₈
Tourism recreation	s₂	s₇	s₆	s₉	s₈	s₉
River traffic	s₄	s₇	s₅	s₈	s₇	s₈

Table 6.

LPCM with respect to national income L21

Regional gains	Electric power generation	Irrigation	Flood protection	Water supply	Tourism recreation	River traffic
Electric power generation	s₈	s₄	s₆	s₃	s₆	s₈
Irrigation	s₁₂	s₈	s₉	s₄	s₉	s₁₁
Flood protection	s₁₀	s₇	s₈	s₈	s₉	s₁₀
Water supply	s₁₃	s₁₂	s₈	s₈	s₈	s₁₄
Tourism recreation	s₁₀	s₇	s₇	s₈	s₈	s₁₂
River traffic	s₈	s₅	s₆	s₂	s₄	s₈

Table 7.

LPCM with respect to regional gains L22

Balance of payment	Electric power generation	Irrigation	Flood protection	Water supply	Tourism recreation	River traffic
Electric power generation	s₈	s₁₀	s₁₄	s₁₃	s₁₀	s₁₁
Irrigation	s₆	s₈	s₁₂	s₉	s₁₀	s₇
Flood protection	s₂	s₄	s₈	s₅	s₂	s₆
Water supply	s₃	s₇	s₁₁	s₈	s₇	s₉
Tourism recreation	s₆	s₆	s₁₄	s₉	s₈	s₉
River traffic	s₅	s₉	s₁₀	s₇	s₇	s₈

Table 8.

LPCM with respect to balance of payment L23

Import substitution	Electric power generation	Irrigation	Flood protection	Water supply	Tourism recreation	River traffic
Electric power generation	s₈	s₁₀	s₁₆	s₁₄	s₁₁	s₁₀
Irrigation	s₆	s₈	s₁₀	s₁₃	s₉	s₆
Flood protection	s₀	s₆	s₈	s₇	s₅	s₄
Water supply	s₂	s₃	s₉	s₈	s₃	s₃
Tourism recreation	s₅	s₇	s₁₁	s₁₃	s₈	s₇
River traffic	s₆	s₁₀	s₁₂	s₁₃	s₉	s₈

Table 9.

LPCM with respect to import substitution L24

Foreign exchange	Electric power generation	Irrigation	Flood protection	Water supply	Tourism recreation	River traffic
Electric power generation	s₈	s₁₁	s₁₃	s₁₄	s₉	s₉
Irrigation	s₅	s₈	s₉	s₉	s₈	s₆
Flood protection	s₃	s₇	s₈	s₉	s₃	s₈
Water supply	s₂	s₇	s₇	s₈	s₄	s₂
Tourism recreation	s₇	s₈	s₁₃	s₁₂	s₈	s₈
River traffic	s₇	s₁₀	s₈	s₁₄	s₈	s₈

Table 10.

LPCM with respect to foreign exchange L25

Firstly, we use model (14) to obtain the optimal objective value M^*, and we further utilize model (20) to obtain the unique individual numerical scale of S^AHP, which are listed in Table 11.

s_i	s₀	s₁	s₂	s₃	s₄	s₅	s₆	s₇	s₈
NS^AHP (s_i)	0.11	0.12	0.15	0.18	0.22	0.27	0.38	0.62	1

s_i	s₉	s₁₀	s₁₁	s₁₂	s₁₃	s₁₄	s₁₅	s₁₆
NS^AHP (s_i)	1.6	2.6	3.66	4.66	5.6	6.6	8.01	9.4

Table 11.

The individual numerical scales

Meanwhile, the LPCMs are transformed into NPCMs using the obtained numerical scales with the PIS. The transformed NPCMs are listed below:

A11=(11.64.62.61.60.6216.62.62.60.220.1510.270.220.380.383.6612.60.620.384.60.381 ),

A21=(14.62.65.66.64.60.2210.150.621.61.60.386.616.62.63.660.181.60.1510.6210.150.620.381.611.60.220.620.2710.621 ),

A22=(10.220.380.180.3814.611.60.221.63.662.60.62111.62.65.64.61116.62.60.620.62114.610.270.380.150.221 ),

A23=(12.66.65.60.623.660.3814.61.60.620.620.150.2210.270.150.380.180.623.6610.621.60.380.386.61.611.60.271.60.620.620.621 ),

A24=(12.69.46.63.662.60.3812.65.61.60.380.110.3810.270.270.220.150.181.610.180.180.270.623.665.610.620.382.64.65.61.61 ),

A25=(13.665.66.61.61.60.2711.61.610.380.180.6211.60.1810.150.620.6210.220.150.6215.64.6110.622.616.611 ).

From NPCMs A11 , A21 , A22 , A23 , A24 and A25 , the local priority vector Pkz(z=1,2,...,nk, k=1,2,...,m−1) can be yielded using model (1), which listed in the following:

P11=(0.3245,0.3101,0.0458,0.1799,0.1397)T,P21=(0.4272,0.0759,0.2878,0.0662,0.0802,0.0623)T,P22=(0.3391,0.1097,0.0820,0.0460,0.2201,0.2031)T,P23=(0.0532,0.1960,0.1847,0.3441,0.1731,0.0490)T,P24=(0.4045,0.1514,0.0405,0.0413,0.1374,0.2249)T,P25=(0.3989,0.1791,0.0.366,0.1113,0.1535,0.1206)T.

Subsequently, a global priority vector can be generated by synthesizing the local priority vectors using Eq. (4). The global priority vector of alternatives is listed below:

w=(0.3787,0.1186,0.14,0.0739,0.1474,0.1413)T

Then the ranking of alternatives can be derived from w : Electric power generation > Tourism recreation > River traffic > Flood protection > Irrigation > Water supply.

6. Simulation and comparison analysis

In this section, we present simulation and comparison analysis to discuss the validity of our proposal. Specifically, the comparison criteria are presented. Then, the simulation method is designed. Finally, the comparison results are analyzed.

6.1. Comparison criteria

The consistency is the basics of the preference relations, and we hope that the NPCMs transformed from the corresponding LPCMs are as consistent as possible. Thus, the consistency index of the transformed NPCM is an important criterion to evaluate the decision efficiency of the AHP model. The Saaty’s consistency index and geometric consistency index have been widely used in the AHP. Therefore, we use these two consistency indexes to measure the consistency level of transformed NPCM. Moreover, the deviation between a NPCM and its priority vector should be as smaller as possible. Thus, we define the deviation between a NPCM and its priority vector as a comparison criterion. The Euclidean distance and Manhattan distance have been widely used in decision making models, so we use these two distance measures to compute the deviation between a NPCM and its priority vector. Saaty’s consistency index has been introduced in Section 2. In the following, we present the geometric consistency index, Euclidean distance between a NPCM and its priority vector, and Manhattan distance between a NPCM and its priority vector.

(1) Geometric consistency index

Let A = (a_ij)_n×n be a NPCM, and P = (p₁,…, p_n)^T be the priority vector obtained from A. Crawford and Williams [9] proposed the geometric consistency index (GCI) for measuring the consistency level of the NPCM A = (a_ij)_n×n:

(21)GCI(A)=2(n−1)(n−2)∑i<j(log(aij)+log(pj)−log(pi))2

Clearly, GCI(A) ≥ 0. GCI(A) = 0 indicates that the NPCM A is a completely consistent NPCM. The smaller GCI(A) value signifies the better consistency level of the NPCM A.

(2) Euclidean distance between a NPCM and its priority vector

The Euclidean distance [10], ED, between the NPCM A and its priority vector P is computed by:

(22)ED(A)=1n∑j=1n∑i=1n(aij−pipj)2

Clearly, ED(A) ≥ 0. ED(A) = 0 indicates that the priority vector P is completely consistent with the NPCM A. The smaller ED(A) value indicates the better efficiency of the AHP model.

(3) Manhattan distance between a NPCM and its priority vector

The Manhattan distance, MD, between A and P is calculated by:

(23)MD(A)=1n2∑i=1n∑j=1n|log(aij)−log(pipj)|

Clearly, MD(A) ≥ 0. MD(A) = 0 indicates that the priority vector P is completely consistent with the NPCM A. The smaller MD(A) value indicates the better efficiency of the AHP model.

6.2. Simulation method

In this section, we design simulation experiment to compare our proposal with the AHP with a fixed numerical scale (FNS) (i.e., the Saaty’s numerical scale). The basic idea of the simulation method is as follows:

We randomly and uniformly generate the LPCMs on a given hierarchical structure based on the linguistic term set S^AHP = {s₀, s₁,…, s₁₆}. Then, we take the generated LPCMs as the inputs of our proposal (i.e., the AHP with the PIS) and the AHP with the FNS, and we can obtain the average values of CR, GCI, ED, and MD for all NPCMs (i.e., ACR, AGCI, AED, and AMD) under these two approaches, respectively.

The simulation method is presented in Table 12.

Input: m, n_k (k = 1,…,m − 1), nkz (z=1,…,nk; k=1,…,m−1)

Output: ACR^(PIS), AGCI^(PIS), AED^(PIS), AMD^(PIS), ACR^(FNS), AGCI^(FNS), AED^(FNS), AMD^(FNS) values.

Step 1: Based on the established hierarchical structure, we generate ∑k−1m−1nk LPCMs Lkz=(lijk,z)nkz×nkz (z=1,…,nk; k=1,…,m−1) with weak stochastic transitivity, where lijk,z is uniformly and randomly selected from S^AHP.

Step 2: Taking the generated LPCMs Lkz=(lijk,z)nkz×nkz (z=1,...,nk;k=1,...,m−1) as the inputs of model (14), we can obtain the optimal objective value M^*. Then, model (20) is used to find the unique individual numerical scales NS^AHP (s_i) (i = 0,1,2,…,16). Following this, we can transform the LPCMs Lkz into NPCMs Akz=(aijk,z)nkz×nkz using the obtained individual numerical scales NS^AHP (s_i) (i = 0,1,2,…,16). Subsequently, we utilize model (1) to yield the priority vector Pkz from the corresponding NPCM Akz . Meanwhile, CR(Akz) , GCI(Akz) , ED(Akz) , MD(Akz) can be obtained. Finally, we calculate the average value of CR(Akz) , GCI(Akz) , ED(Akz) , MD(Akz) for all NPCMs, that are

ACR(PIS)=∑k−1m−1∑z−1nkCR(Akz)/∑k=1m−1nk,AGCI(PIS)=∑k=1m−1∑z=1nkGCI(Akz)/∑k=1m−1nk,AED(PIS)=∑k=1m−1∑z=1nkED(Akz)/∑k−1m−1nk,AMD(PIS)=∑k=1m−1∑z=1nkMD(Akz)/∑k=1m−1nk.

Step 3: Using the fixed numerical scale, NSAHP(si)={1/(9−i),i=0,1,…,8i−7,i=9,10,…,16 , transforms the LPCMs Lkz into NPCMs Akz . Subsequently, utilize model (1) to yield the priority vector Pkz from NPCM Akz . Meanwhile, CR(Akz) , GCI(Akz) , ED(Akz) , MD(Akz) can be obtained. Finally, we calculate the average values of CR(Akz) , GCI(Akz) , ED(Akz) , MD(Akz) , for all NPCMs, that are

ACR(FNS)=∑k−1m−1∑z−1nkCR(Akz)/∑k−1m−1nk,AGCI(FNS)=∑k=1m−1∑z=1nkGCI(Akz)/∑k=1m−1nk,AED(FNS)=∑k=1m−1∑z=1nkED(Akz)/∑k=1m−1nk,AMD(FNS)=∑k=1m−1∑z=1nkMD(Akz)/∑k=1m−1nk.

Step 4: Output ACR^(PIS), AGCI^(PIS), AED^(PIS), AMD^(PIS), ACR^(FNS), AGCI^(PIS), AED^(FNS), and AMD^(FNS).

Table 12.

Simulation method

6.3. Comparison results

In this section, we set different parameter values for the simulation method to analyze the performance of our proposal.

We consider a hierarchical structure with three levels: goal level, criterion level and alternative level. In this case, m = 3 and n₁ = 1. Let ∆ = 0.4, when setting different n₂ and n₃ values as the inputs of the simulation method, and we run it 1000 times to generate the average values of ACR, AGCI, AED, and AMD under our proposed approach (i.e., AHP with the PIS) and the existing AHP approach (i.e., AHP with the FNS), respectively, which are described in Figs. 5-8.

From Figs. 5–8, we have the following observations.

The Saaty’s consistency index and geometric consistency index of the transformed NPCM, the Manhattan distance and Euclidean distance between a NPCM and its priority vector in the AHP with the PIS are obviously lower than those in the AHP with the FNS, respectively. This finding implies that the proposed approach can improve the decision efficiency by taking the PIS into account.

7. Conclusion

In the AHP, the PIS is an important factor that cannot be ignored due to its influence on the final decision results. This study proposes an AHP framework based PIS for supporting the decision making. The main work of this paper is summarized as follows:

(1)
For implementing the computation process, in the AHP the linguistic preference information is often transformed into the numerical preference information using a fixed numerical scale function. In other words, the PIS is not considered by the existing AHP framework. By taking the PIS into account, this study proposes a novel AHP framework, which is closer to the realistic decision scenario.
(2)
We develop a consistency-driven approach that minimizes the inconsistency level of the transformed numerical preference information (i.e., multiplicative preference relations) to support the proposed AHP framework. In the consistency -driven approach, a two-stage based optimization model is designed to deal with the PIS, and we transform it into linear programming model that can be easily solved.
(3)
We design the detailed simulation experiment to verify the validity of the AHP framework with the PIS, and the simulation results show that our proposal has a better decision efficiency compared with the existing AHP framework.

Meanwhile, the following research paths are pointed out for the future:

(1)
In our proposal, each element in an LPCM is mapped to an exact number. However, an LPCM is usually with uncertainty. To deal with this issue, the uncertain numerical scale (interval numerical scale) has been proposed and used in the linguistic decision making [19], further studies should discuss the uses of the interval numerical scale in the PIS based AHP framework.
(2)
Practical decision making involves both mathematical models and psychological issues (e.g., prospect theory [17, 36]). However, psychological issues are seldom considered in the existing AHP models. To study the psychological issues of decision maker(s) in the PIS based AHP framework is a very interesting research direction.
(3)
In addition, the consensus based group decision making has become a hot topic [11, 18, 20]. It would be interesting in any future research to export the PIS based AHP framework to the consensus based group decision making.

Acknowledgements

This work was supported by the grant (No. 71571124) from NSF of China, and the grant (No. 2017B07514) from “the Fundamental Research Funds for the Central Universities”.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 11 - 1
Pages: 451 - 468
Publication Date: 2018/01/01
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.11.1.34 How to use a DOI?
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TY  - JOUR
AU  - Qiuxiang Zhou
AU  - Yucheng Dong
AU  - Hengjie Zhang
AU  - Yuan Gao
PY  - 2018
DA  - 2018/01/01
TI  - The analytic hierarchy process with personalized individual semantics
JO  - International Journal of Computational Intelligence Systems
SP  - 451
EP  - 468
VL  - 11
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.11.1.34
DO  - 10.2991/ijcis.11.1.34
ID  - Zhou2018
ER  -

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