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European Journal of Operational Research 186 (2008) 229–242 www.elsevier.com/locate/ejor

Decision Support

A comparative study of the numerical scales and the prioritization methods in AHP Yucheng Dong a

a,*

, Yinfeng Xu

a,b

, Hongyi Li c, Min Dai

a

Department of Management Science, Management School, Xi’an Jiaotong University, Xi’an 710049, China b State Key Laboratory for Manufacturing Systems Engineering, Xi’an 710049, China c Faculty of Business Administration, The Chinese University of Hong Kong, Shatin, NT, Hong Kong Received 20 February 2006; accepted 10 January 2007 Available online 7 March 2007

Abstract In the analytic hierarchy process (AHP), a decision maker ﬁrst gives linguistic pairwise comparisons, then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them, and ﬁnally derives a priority vector from the numerical pairwise comparisons. In particular, the validity of this decision-making tool relies on the choice of numerical scale and the design of prioritization method. By introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices (LPCMs), and by deﬁning the deviation measures of LPCMs, we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods. Using these performance measure algorithms, we compare the most common numerical scales (the Saaty scale, the geometrical scale, the Ma–Zheng scale and the Salo–Ha¨ma¨la¨inen scale) and the prioritization methods (the eigenvalue method and the logarithmic least squares method). In addition, we also discuss the parameter of the geometrical scale, develop a new prioritization method, and construct an optimization model to select the appropriate numerical scales for the AHP decision makers. The ﬁndings in this paper can help the AHP decision makers select suitable numerical scales and prioritization methods. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Decision analysis; AHP; Linguistic pairwise comparison matrices; Numerical scale; Prioritization method

1. Introduction To our knowledge, the pairwise comparison method originates from psychological research [33]. Saaty improved it mathematically and made it the basis of the analytic hierarchy process (AHP) *

Corresponding author. Tel.: +86 2982673492. E-mail addresses: [email protected] (Y. Dong), yfxu@ mail.xjtu.edu.cn (Y. Xu), [email protected] (H. Li), [email protected] (M. Dai).

[22,23,26]. In the last 20 years, AHP has been used in almost all the applications related to multiple criteria decision-making [35,36]. But there is still much dispute on this decision-making tool [3,7,8,11,21,24]. In AHP, a decision maker ﬁrst gives linguistic pairwise comparisons, then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them, and ﬁnally derives a priority vector from the numerical pairwise comparisons. In particular, the validity of this decision-making tool relies on the following factors.

0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.01.044

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(1) Selection of numerical scale. This is the selection of a reasonable numerical scale to quantify the linguistic pairwise comparisons and thereby to obtain the numerical pairwise comparison matrix. There have been several diﬀerent numerical scales such as the Saaty scale [22], the Ma-Zheng scale [20], the geometrical scale [10,17,19] and the Salo-Ha¨ma¨la¨inen scale [30]. The Saaty scale and the geometrical scale are the most commonly used ones. The Saaty scale has been supported by Saaty’s empirical evidence [22], but it is not a transitive scale. The geometrical scale is thought to be transitive, however, Saaty [27] points out that it is diﬃcult to determine the geometrical scale parameter. (2) Design of prioritization method. The prioritization method refers to the process of deriving a priority vector from the numerical pairwise comparison matrix. There have been a large number of prioritization methods [4,31], among which the eigenvalue method (EVM) [22] and the logarithmic least squares method (LLSM) [5] are most commonly used. There is some dispute on which method is better. The literatures [2,5,32,41] argue that LLSM is better, while the Saaty school [18,25,28,29] holds an opposite opinion. The main purpose of this paper is to compare several common numerical scales (the Saaty scale, the geometrical scale, the Ma-Zheng scale and the Salo–Ha¨ma¨la¨inen scale) and prioritization methods (the EVM and the LLSM). The paper is organized as follows. In Section 2, we introduce the notations and the operational laws of the linguistic variables and the linguistic pairwise comparison matrices (LPCMs), which will be the mathematical basis for our comparative study. In Section 3, we formally present two performance measure algorithms to evaluate the numerical scales and the prioritization methods. In Section 4, using these two performance measure algorithms, we compare the numerical scales and the prioritization methods. In Section 5, we discuss some interrelated issues on scale and prioritization, including the parameter of the geometrical scale. A new prioritization method, and an optimization model to select appropriate numerical scales for the AHP decision makers are also discussed. Finally, the concluding remarks are presented in Section 6.

2. Preliminary knowledge 2.1. Linguistic pairwise comparison matrices The basic notations and operational laws of the linguistic variables and LPCMs (i.e., the linguistic preference relations) have been widely discussed [6,9,14–16,38,39]. Let S ¼ fsa j a ¼ t; . . . ; 1; 0; 1; . . . ; tg be a linguistic label set with odd cardinality. A label sa represents a possible value for a linguistic variable. It is required that the linguistic label set should satisfy the following characteristics: (1) The set is ordered: sa > sb if and only if a > b; (2) There is a negation operator: neg ðsa Þ ¼ sa . We call this linguistic label set S the linguistic scale. For example, S can be deﬁned as: S ¼ fs4 ¼ extremely poor

s3 ¼ very poor;

s2 ¼ poor s1 ¼ slightly poor; s0 ¼ fair; s1 ¼ slightly good s2 ¼ good; s3 ¼ very good; s4 ¼ extremely goodg: To preserve all the given information, we extend the discrete linguistic label set S to a continuous linguistic label set S ¼ fsa j a 2 ½q; qg, where qðq P tÞ is a suﬃciently large positive real number. If sa 2 S, we call sa the original linguistic label; if sa 2 S, then we call it the virtual linguistic label. In general, the decision maker uses the original linguistic labels to evaluate the alternatives, and the virtual linguistic labels can only appear in operations. Consider any two linguistic terms sa ; sb 2 S, and l; l1 ; l2 2 ½0; 1. Xu [38,39] introduces the following operational laws: ð1Þ sa sb ¼ saþb ; ð2Þ sa sb ¼ sb sa ; ð3Þ lsa ¼ sla ; ð4Þ ðl1 þ l2 Þsa ¼ l1 sa l2 sa ; ð5Þ lðsa sb Þ ¼ lsa lsb . Let X ¼ fx1 ; x2 ; . . . ; xn gðn P 2Þ be a ﬁnite set of alternatives. When a decision maker makes pairwise comparisons using the linguistic label set S, he/she can construct a LPCM to represent his/her own opinion on X. The LPCM can be formally deﬁned as follows. Deﬁnition 1. The matrix D ¼ ðd ij Þnn is called a discrete linguistic pairwise comparison matrix (LPCM) if d ij 2 S and d ij d ji ¼ s0 ; ði; j ¼ 1; 2; . . . ; nÞ. Deﬁnition 2. The matrix D ¼ ðd ij Þnn is called a continuous linguistic pairwise comparison matrix

Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

(LPCM) if d ij 2 S and d ij d ji ¼ s0 ; ði; j ¼ 1; 2; . . . ; nÞ. Generally, the LPCMs constructed by decision makers are discrete, and the continuous LPCMs can only appear in operations. Next, let sa ; sb 2 S be two linguistic variables. Xu [39] deﬁnes the distance (i.e., the deviation degree) between sa and sb as follows: dðsa ; sb Þ ¼

ja bj ; T

ð1Þ

where T is the number of linguistic terms in the set S. If only one pre-established linguistic label set is used in a decision making model, we can simply consider dðsa ; sb Þ ¼ ja bj. Xu [39] further discusses the deviation degree between two LPCMs. Let A ¼ ðaij Þnn and B ¼ ðbij Þnn be two LPCMs, Dong and Xu [6] propose the following deviation measure between A and B: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u n n X X u 2 2 ð2Þ dðA; BÞ ¼ t ðdðaij ; bij ÞÞ : nðn 1Þ j¼iþ1 i¼1 The properties of the deviation measures between LPCMs are discussed in [6,39]. In AHP, there is a pre-established linguistic label set (i.e., the AHP linguistic scale) [10,17,22] S AHP ¼ fs8 ¼ extremely less important; s7 ¼ very; very strongly less important; s6 ¼ demonstratedly less important; s5 ¼ strongly plus less important; s4 ¼ strongly less important; s3 ¼ moderately plus less important; s2 ¼ moderately less important; s1 ¼ weakly less important;

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using SAHP, and then construct discrete LPCMs to represent their opinions. Note: In this paper, we introduce the basic notations and operational laws of the linguistic variables and the LPCMs into AHP. This enable us to apply the existing knowledge about linguistic approaches to the AHP discipline. These newly introduced concepts have some similarity to the idea of digitized verbal part presented in Ji and Jiang [17]. Ji and Jiang [17] adopt digitized verbal part to express the AHP linguistic scale, and study the scale transitivity in details. 2.2. Scale function and prioritization method Let s 2 S. We denote IðsÞ the lower indices of s, and call it the scale gradation of s in S. For example, if s ¼ sa , then IðsÞ ¼ a. Let D ¼ ðd ij Þnn be a LPCM. If d ij ¼ sa , then Iðd ij Þ ¼ a. Deﬁnition 3. A monotonically increasing function f: S AHP ! Rþ is called a scale function if f ðsa Þ f ðsa Þ ¼ 1, where S AHP ¼ fsa j a 2 ½q; qg and f ðsq Þ > 0. Harker and Vargas [12] argue that the choice of a numerical scale is an open research issue. In fact, this problem of setting a suitable numerical scale in AHP is equivalent to looking for an appropriate scale function. Several common numerical scales and their scale functions are listed below. (1) The saaty scale [22] ( IðsÞ þ 1 s P s0 ; f ðsÞ ¼ 1 s < s0 : 1IðsÞ (2) The geometrical scale pﬃﬃﬃ f ðsÞ ¼ ð cÞIðsÞ :

s0 ¼ equally important; s1 ¼ weakly more important; s2 ¼ moderately more important; s3 ¼ moderately plus more important; s4 ¼ strongly more important; s5 ¼ strongly plus more important; s6 ¼ demonstratedly more important; s7 ¼ very; very strongly more important; s8 ¼ extremely more importantg: According to the decision process of AHP, the AHP decision makers ﬁrst make pairwise comparisons

The geometrical scale is thought to be transitive. Some researchers examine its transitivity recently [10,17]. For the geometrical scale parameter, Lootsma [19] suggests to take values of 2 or 4. Finan and Hurley [10] argue that a reasonable value ranges between 1.2 and 2. Ji and Jiang [17] present interesting methods to determine the scale parameter, and show that the value is close to 2. (3) The Ma-Zheng scale [20] ( 9 s P s0 ; 9IðsÞ f ðsÞ ¼ 9þIðsÞ s < s0 : 9

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(4) The Salo-Ha¨ma¨la¨inen Scale [30] f ðsÞ ¼

0:5 þ IðsÞ e ; 0:5 IðsÞ e

where e ¼ 201 or e ¼ 171 . Let D be a discrete LPCM constructed by the AHP decision maker. By selecting a suitable scale function f to quantify D, we obtain a numerical pairwise comparison matrix A ¼ ðaij Þnn , where aij ¼ f ðd ij Þ; i; j ¼ 1; 2; . . . ; n. The prioritization method refers to the process of deriving the priority vector from the numerical pairwise comparison matrix. The two most common prioritization methods (EVM and LLSM) [4,5,22,31] are listed below. (1) The Eigenvalue Method Saaty [22] proposes the principal eigenvector of A as the desired priority vector w, which can be obtained by solving the linear system: Aw ¼ kw;

eT w ¼ 1;

where k is the principal eigenvalue of matrix A. (2) The Logarithmic Least Squares Method The LLSM method uses the L2 metric in deﬁning an objective function of the following optimization problem: 8 n P P 2 > ½lnðaij Þ ðlnðwi Þ lnðwj ÞÞ ; > < min i¼1 j>i

> > : s:t:

wi P 0;

n P

wi ¼ 1:

i¼1

Crawford and Williams [5] have shown that the solution to the above problem is unique and can be found simply as the geometric means of the rows of matrix A: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n Q 1=n aij wi ¼

j¼1

Pn

i¼1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! : n Q aij

1=n

scale, namely the scale T function. For example, consider that w ¼ 23 ; 13 is a priority vector whose characteristic matrix is 1 2 wi W ¼ ðwij Þ22 ¼ ¼ 1 : wj 22 1 2 If the scale function selected by the AHP decision maker is the Saaty scale, the priority vector signiﬁes that alternative 1 is weakly more important than alternative 2 (w1;2 ¼ 2 and the corresponding linguistic scale is s1); however, if the selected scale function is the geometrical scale ðc ¼ 2Þ, then it signiﬁes that alternative 1 is moderately more important than alternative 2 (the corresponding linguistic scale is s2). Therefore, the priority vector is the same, but with diﬀerent semantics because of the diﬀerent scale functions selected. Let D ¼ ðd ij Þnn ¼ ðf 1 ðwwji ÞÞnn . Denote F ¼ ffa j fa ¼ f ðsa Þ; sa 2 S AHP g as the discrete numerical scale values of f. Theorem 1. (1) D ¼ ðd ij Þnn , where d ij ¼ f 1 ðwwji Þ, is a continuous LPCM; (2) d ij ¼ sa if wwji ¼ fa ; (3) sa < d ij < saþ1 if fa < wwji < faþ1 . Proof. Let x ¼ f ðsa Þ. Because f ðsa Þ f ðsa Þ ¼ 1, we have f ðsa Þ ¼ 1x. Because f 1 ðf ðsa ÞÞ f 1 ðf ðsa ÞÞ ¼ s0 , it can be shown that 1 1 1 ð3Þ ¼ s0 : f ðxÞ f x Consequently, 1 wi 1 wj f f ¼ s0 : wj wi

ð4Þ

Based on Deﬁnition 1, D is a continuous LPCM. Since fa ¼ f ðsa Þ, where sa 2 S AHP , we have that wi 1 wi d ij ¼ f ¼ fa : ð5Þ ¼ sa ; if wj wj Since f 1 ðxÞ is a monotonically increasing function, we further obtain that

j¼1

sa < d ij < saþ1 ; 2.3. LPCM of the priority vector

if

fa
wj > > > > sa fa < wwji < faþ1 and > > > > > > d f 1 wwji ; sa < d f 1 wwji ; saþ1 ; > > < wi d ij ¼ saþ1 fa < wj < faþ1 and > > > > d f 1 wwji ; saþ1 6 d f 1 wwji ; sa : > > > > > > s8 f ðwwji Þ 6 s8 ; > > > : s8 f ðwwji Þ P s8 : ð10Þ This completes the proof of Theorem 2. h We call D the discrete LPCM of the priority vector w. 3. Two performance measure algorithms As stated in previous sections, the decision maker ﬁrst gives linguistic pairwise comparisons in AHP,

233

then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them, and ﬁnally derives a priority vector from the numerical pairwise comparisons. From the decision process steps of AHP, it is easy to know that the input of AHP is a LPCM given by a decision maker, and the output is a priority vector. The validity of this decision tool lies on the choice of numerical scale and the design of prioritization method. In the rest of this section, we will present two performance measure algorithms to evaluate the numerical scales and the prioritization methods considered. The essence of these two performance measure algorithms is to calculate the diﬀerence between input and output of AHP. Obviously, the smaller the diﬀerence, the better the selected numerical scale and prioritization method. However, the input and the output have diﬀerent representation format (i.e., LPCM and the priority vector) in AHP. Therefore, the key of designing these two algorithms is to unify the format of input and output by extending the decision process steps of AHP. 3.1. Performance measure Algorithm I Theorem 1 shows that a continuous LPCM of the priority vector can explain the semantics of the corresponding priority vector. Thus, we can transform the derived priority vector into a LPCM. By computing the diﬀerence between the initial LPCM given by a decision maker and the LPCM of the derived priority vector, we design the ﬁrst performance measure algorithm. Algorithm I Input: The LPCM, D ¼ ðd ij Þnn , given by a decision maker. Output: The continuous LPCM of the derived priority vector D ¼ ðd ij Þnn , and the deviation degree between D and D, namely dðD; DÞ. Step 1: Let D ¼ ðd ij Þnn be a LPCM constructed by the AHP decision maker. Step 2: Use certain scale function f ðxÞ to quantify D ¼ ðd ij Þnn , and thereby obtain a numerical pairwise comparison matrix A ¼ ðaij Þnn , where aij ¼ f ðd ij Þ. T Step 3: Let w ¼ ðw1 ; w2 ; . . . ; wn Þ be the priority vector derived from A by certain prioritization method. Step 4: Obtain the continuous LPCM of the priority vector, D ¼ ðd ij Þnn , where d ij ¼ f 1 wwji . Step 5: Calculate dðD; DÞ.

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dðD; DÞ denotes the deviation degree between an initial LPCM given by a decision maker and the continuous LPCM of the derived priority vector. We propose to use this as an index to evaluate the numerical scales (or scale functions) and the prioritization methods considered. This index has a deﬁnite physical implication. It reﬂects the diﬀerence between the semantics of the priority vector and the initial semantics given by a decision maker. We try to ﬁnd suitable scale and prioritization that make the value of dðD; DÞ as small as possible. In this paper, we use formula (2) to calculate the deviation degree between D and D, namely vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u n n X X u 2 2 ð11Þ dðD; DÞ ¼ t ðdðd ij ; d ij ÞÞ : nðn 1Þ j¼iþ1 i¼1

3.2. Performance measure Algorithm II In [22], Saaty introduces four examples (the Distance Problem, the Optics Problem, the Nation Wealth Problem and the Weight Estimation Problem). The true priority vector (w) is known and the initial LPCM (D) has been provided by the decision maker. Using certain scale and prioritization, the derived priority vector ðwÞ can be obtained. Saaty adopts these real examples to evaluate the numerical scales by computing the root-meansquare deviation (RMS) between w and w: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uP un u ðwi wi Þ2 t RMS ¼ i¼1 : n Here, we further extend Saaty’s method by assuming that the AHP decision maker is rational. If T w ¼ ðw1 ; w2 ; . . . ; wn Þ is a true priority vector and the scale function adopted by the decision maker is f ðxÞ, then the continuous LPCMs of the truepri ority vector is D ¼ ðd ij Þnn , where d ij ¼ f 1 wwji . Because the decision maker can only choose discrete linguistic variables from SAHP in making pairwise comparisons, a rational decision maker will naturally choose the closest linguistic variables from SAHP to approximate the continuous LPCM of the true priority vector, D ¼ ðd ij Þnn . That is to say, a rational decision maker will adopt the discrete LPCM of the true priority vector to approximate D. Based on this, we present the performance measure Algorithm II.

Algorithm II Input: The true priority vector w. Output: The derived priority vector w, and the deviation degree between w and w, namely RMS. Step 1: Let w ¼ ðw1 ; w2 ; . . . ; wn ÞT be a true priority vector. Step 2: Let f ðxÞ be the scale function adopted by the decision maker. Denote D ¼ ðd ij Þnn , where d ij ¼ f 1 ðwwji Þ, as the continuous LPCM of the priority vector. Step 3: Assume that the decision maker is rational, and he/she always chooses the closest linguistic variable d ij from SAHP to approximate d ij in making pairwise comparisons. Denote D ¼ ðd ij Þnn as the approximated linguistic matrix. Step 4: Obtain the numerical pairwise comparison matrix A ¼ ðaij Þnn , where aij ¼ f ðd ij Þ. Step 5: Derive the priority vector w ¼ ðw1 ; w2 ; T . . . ; wn Þ from A by certain prioritization method. Step 6: Compute the root-mean-square ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ deviation rP n ðwi wi Þ2 i¼1 (RMS), where RMS ¼ . n

Obviously, the smaller the RMS, the better the selected numerical scale and the prioritization method. Remark 1. The performance measure Algorithm II is inspired by Saaty’s method. In Saaty’s method, both the true priority vectors and the LPCMs provided by the decision makers are used as the inputs. For the same problems, diﬀerent decision makers may provide diﬀerent LPCMs. Thus the evaluation results of Saaty’s method depend on the individual decision makers. Algorithm II is such a method without the interventions of the decision makers. The algorithm assumes that the decision makers are rational, and it only uses priority vectors as inputs, which is the diﬀerence between these two methods. Remark 2. Algorithm II is also inspired by the simulation method presented by Triantaphyllou and Mann [34]. In [34], the AHP decision makers are assumed to be rational as well. They choose the closest values from f19 ; 18 ; . . . ; 1; . . . ; 8; 9g (take the Saaty scale as an example) to approximate the characteristic matrix of the true priority vector

Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

W ¼ ðwij Þnn ¼ ðwwji Þnn . However, this method sometimes will cause the approximated matrix W to lose the reciprocal property. For example, given T a true priority vector w ¼ ð3:45; 1Þ . Its characteristic matrix is ! 1 3:45 W ¼ : 1 1 3:45

wð1Þ ¼ ð0:2777; 0:3611; 0:0320; 0:1324; 0:1773; 0:0194ÞT :

The LPCM on SAHP is given by 0

Dð1Þ

In the set f1=9; 1=8; . . . ; 1; . . . ; 8; 9g, the closest 1 value to 3.45 is 3 and the closest value to 3:45 is 14, therefore the approximated numerical pairwise comparison matrix is W ¼

1

3

1 4

1

:

Obviously, its reciprocal property is missing. Thus, it’s unreasonable to consider W as the numerical pairwise comparison matrix given by the rational decision makers. Theorem 2 guarantees that D is reciprocal, and also shows the rationality to use D as the LPCM given by the rational decision makers. Besides, the AHP decision makers use the linguistic scale, not the numerical scale to make pairwise comparisons. So it seems to be more reasonable to select the closest linguistic variables from SAHP to approximate the continuous LPCM of the true priority vector.

s0

B B s2 B B B s7 B ¼B B s2 B B B s2 @ s6

s2

s7

s2

s2

s0

s8

s2

s2

s8

s0

s5

s4

s2

s5

s0

s2

s2

s4

s2

s0

s8

s1

s5

s5

s6

1

C s8 C C C s1 C C C: s5 C C C s5 C A s0

Example 2. The Optics Problem. This example deals with the estimation of the relative brightness. The values of the actual relative brightness are T

wð2Þ ¼ ð0:6072; 0:2186; 0:1115; 0:0627Þ : The two LPCMs on SAHP are 0

s0

s4

s5

s6

1

B B s4 Dð2;1Þ ¼ B B @ s5

s0

s3

s3

s0

C s5 C C; C s3 A

s6

s5

s3

s0

s0

s3

s5

s6

0 D

ð2;2Þ

1

B B s3 ¼B B @ s5

s0

s2

s2

s0

C s3 C C: C s1 A

s6

s3

s1

s0

4. Comparison of the numerical scales and prioritization methods In this section, we use both real data and random data as the inputs of our performance measure algorithms to evaluate numerical scales and prioritization methods. The experimental results obtained by our algorithms will also be compared with some existing methods.

235

Example 3. The Nation Wealth Problem. The example deals with the estimation of the relative wealth of seven countries: the United States, the USSR, China, France, the UK, Japan, and West Germany. Their actual GNP fractions are, respectively, wð3Þ ¼ ð0:4134; 0:2249; 0:0425; 0:0694; 0:0546; T

0:1041; 0:0910Þ : 4.1. Experimental results with real data Saaty [22] provides four famous examples to evaluate the numerical scales. Ji and Jiang [17] also follow Saaty’s examples in discussing the parameter of the geometrical scale in details. These four examples are listed below. Example 1. The Distance Problem. The example deals with the distances of Cairo, Tokyo, Chicago, San Francisco, London, and Montreal from Philadelphia. The real relative distances are, respectively,

The LPCM on SAHP is given by 0

Dð3Þ

s0 Bs B 3 B B s8 B B ¼ B s5 B B s5 B B @ s4 s4

s3 s0

s8 s6

s5 s4

s5 s4

s4 s2

s6

s0

s4

s4

s6

s4 s4

s4 s4

s0 s0

s0 s0

s2 s2

s2 s3

s6 s4

s2 s2

s2 s2

s0 s1

1 s4 s3 C C C s4 C C C s2 C: C s2 C C C s1 A s0

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Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

Salo-Ha¨ma¨la¨inen scale e ¼ 201 and e ¼ 171 are better than the Saaty scale (the average values of dðD; DÞ and RMS are smaller). There is not much diﬀerences among the geometrical scale ðc ¼ 2Þ, the Ma-Zheng scale and the Salo-Ha¨ma¨la¨inen scale. (2) When the same scale function is adopted, the diﬀerences between EVM and LLSM are so small that it’s hard to tell the diﬀerences (the average values of dðD; DÞ and RMS are almost the same). This shows that the eﬀects of these two prioritization methods are similar. (3) For the geometrical scale, the average RMS values under the situation of c ¼ 2 are smaller than those when c ¼ 1:5 and c ¼ 4, which shows that the suitable values of c may be close to 2. However, the average values of dðD; DÞ are almost the same when c takes different values.

Example 4 (The Weight Estimation Problem). This example deals with the estimation of the relative weights of ﬁve objects. The actual relative weights are T

wð4Þ ¼ ð0:10; 0:39; 0:20; 0:27; 0:04Þ : The LPCM on SAHP is 0 Dð4Þ

s0

B B s4 B ¼B B s2 B @ s3 s3

s4

s2

s3

s0 s1

s1 s0

s1 s1

s1 s7

s1 s3

s0 s6

s3

1

C s7 C C s3 C C: C s6 A s0

We take Dð1Þ ; Dð2;1Þ ; Dð2;2Þ ; Dð3Þ and Dð4Þ as the inputs of Algorithm I, and then compute the values of dðDðiÞ ; DðiÞ Þði ¼ 1; 2; . . . ; 4Þ and their average values under diﬀerent scales and prioritizations, respectively. The experimental results with real data are shown in Table 1. We also consider wð1Þ ; wð2;1Þ ; wð2;2Þ ; wð3Þ and wð4Þ as the inputs of Algorithm II, and then compute the RMS values and their average RMS values under diﬀerent scales and prioritizations, respectively. The experimental results with real data are shown in Table 2. From Tables 1 and 2, we have the following observations: (1) When the same prioritization method is adopted (EVM or LLSM), the geometrical scale ðc ¼ 2Þ, the Ma-Zheng scale and the

In the rest of the section, we assume that c ¼ 2. In Section 5.1, we will focus on the discussion of the geometrical scale parameter.

4.2. Experimental results with random data In order to make our conclusions more convincing, we will use simulations to further evaluate the numerical scales and the prioritization methods in this subsection. If we generate a large number of LPCMs randomly, then we can calculate the aver-

Table 1 The dðD; DÞ values of Saaty’s examples under diﬀerent scales and prioritizations Scale

Prioritization

Example 1

Example 2 (First Trail)

Example 2 (Second Trail)

Example 3

Example 4

Average values of these examples

Saaty scale

EVM LLSM

1.9600 1.8791

2.6777 2.6700

1.2069 1.1834

2.7661 2.8545

1.9103 1.9582

2.1042 2.1090

Geometrical scale ðc ¼ 1:5Þ

EVM LLSM

0.6531 0.6526

0.7504 0.7500

0.3536 0.3536

0.5455 0.5453

1.8287 1.8199

0.8263 0.8243

Geometrical scale ðc ¼ 2Þ

EVM LLSM

0.6540 0.6526

0.7513 0.7500

0.3536 0.3536

0.5459 0.5453

1.8344 1.8199

0.8278 0.8243

Geometrical scale ðc ¼ 4Þ

EVM LLSM

0.6580 0.6526

0.7548 0.7500

0.3536 0.3536

0.5475 0.5453

1.8470 1.8199

0.8322 0.8243

Ma-Zheng scale

EVM LLSM

1.1711 1.1232

0.2969 0.2981

0.5382 0.5396

0.4687 0.4622

2.0370 2.0136

0.9024 0.8873

Salo scale ðc ¼ 1=20Þ

EVM LLSM

0.7912 0.7879

0.5324 0.5341

0.3985 0.3989

0.4213 0.4228

1.9061 1.8891

0.8099 0.8066

Saloscale ðc ¼ 1=17Þ

EVM LLSM

1.1484 1.0686

0.4428 0.4453

0.4366 0.4376

0.4849 0.4688

1.9741 1.9431

0.8974 0.8727

Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

237

Table 2 The RMS values of Saaty’s examples under diﬀerent scales and prioritizations Scale

Prioritization

Example 1

Example 2

Example 3

Example 4

Average values of these examples

Saaty scale

EVM LLSM

0.0194 0.0202

0.0056 0.0055

0.0088 0.0088

0.0152 0.0148

0.0123 0.0123

Geometrical scale ðc ¼ 1:5Þ

EVM LLSM

0.0255 0.0261

0.0285 0.0295

0.0180 0.0191

0.0142 0.0148

0.0216 0.0224

Geometrical scale ðc ¼ 2Þ

EVM LLSM

0.0033 0.0034

0.0068 0.0068

0.0028 0.0029

0.0047 0.0047

0.0044 0.0045

Geometrical scale ðc ¼ 4Þ

EVM LLSM

0.0277 0.0277

0.0454 0.0454

0.0069 0.0071

0.0161 0.0182

0.0240 0.0246

Ma-Zheng scale

EVM LLSM

0.0110 0.0114

0.0093 0.0093

0.0032 0.0031

0.0077 0.0073

0.0078 0.0078

Salo scale ðc ¼ 1=20Þ

EVM LLSM

0.0150 0.0155

0.0017 0.0017

0.0010 0.0010

0.0062 0.0062

0.0060 0.0061

Salo scale ðc ¼ 1=17Þ

EVM LLSM

0.0127 0.0107

0.0058 0.0058

0.0063 0.0059

0.0105 0.0106

0.0088 0.0083

age values of dðD; DÞ using the performance measure Algorithm I. Obviously the smaller the average values of dðD; DÞ, the better the selected numerical scale and the prioritization option. However, which LPCMs should be selected as computation samples when computing the average values of dðD; DÞ? Naturally, we should select consistent LPCMs as computation samples. The most traditional deﬁnition to characterize the consistency of LPCMs is to use transitivity [31,32]. Thus, in this paper, we randomly select transitive LPCMs. Without loss of generality, we consider that alternative i is preferred to alternative j if i < j. In other words, the entries of the upper triangle of LPCMs are uniformly randomly selected from fsa j a ¼ 0; 1; . . . ; 7; 8g, that is D 2 g, where g ¼ fD ¼ ðd ij Þj d ij d ji ¼ s0

and

d ij P s0 if i < jg: Note: Saaty [22] argues that intransitivity can be allowed in AHP, but it is hard for us to know which intransitive LPCMs can be accepted as our computation samples. Because the judgments of decision makers are generally transitive. Intransitivity is prohibited by most theories [1]. Therefore, we use transitivity as an approximation criterion to select computation samples in this paper. For each of the matrices with diﬀerent sizes, we generate 10000 random LPCMs that belong to g, respectively. By Algorithm I, we can compute the average values of dðD; DÞ under diﬀerent scales and prioritizations. The results with random data are shown in Table 3.

For each diﬀerent priority size, we generate 10000 random priority vectors, w ¼ ðw1 ; w2 ; . . . ; wn ÞT , where wi ði ¼ 1; 2; . . . ; nÞ is uniformly distributed on [0,1]. By Algorithm II, we can obtain the average RMS values under diﬀerent scales and prioritizations. The results are shown in Table 4. Generally speaking, we obtain the same observations as those of the experimental results using Saaty’s real data, which make our conclusions more reliable. 4.3. Summary The performance measure Algorithm II is inspired by Saaty’s method. Using Saaty’s method and examples, we can also compute the RMS values and the average RSM values under diﬀerent scales and prioritizations. The results are showed in Table 5. From Table 5, we obtain diﬀerent observations based on our results by Algorithms I and II. Comparing our results, Saaty’s results and other researchers’ results [20,37,40], we further classify these scales into three types: (1) The geometrical scale ðc ¼ 2Þ. This scale is supported by our performance measure algorithms, Saaty’s method and many other researchers’ results [37,40]. (2) The Saaty scale. The Saaty scale has only been supported by the empirical evidence obtained by Saaty’s method. However, from Table 5,

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Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

Table 3 The average dðD; DÞ values of transitive LPCMs under diﬀerent scales and prioritizations Size of matrix 3 4 5 6 7 8 9

Saaty scale

Geometrical scale ðc ¼ 2Þ

Ma-Zheng scale

Salo scale ðe ¼ 1=20Þ

Salo scale ðe ¼ 1=17Þ

EVM

LLSM

EVM

LLSM

EVM

LLSM

EVM

LLSM

EVM

LLSM

2.0835 2.9213 3.2994 3.4883 3.5987 3.6847 3.7333

2.0670 2.8227 3.1791 3.3789 3.5060 3.5949 3.6419

1.3300 2.0292 2.3981 2.6217 2.7662 2.8741 2.9558

1.3287 1.9843 2.3316 2.5377 2.6821 2.7856 2.8663

1.4737 2.1295 2.4835 2.6756 2.8119 2.9035 2.9614

1.4598 2.0760 2.3761 2.5507 2.6709 2.7480 2.8173

1.3178 1.9907 2.3409 2.5591 2.7046 2.7990 2.8812

1.3121 1.9400 2.2766 2.4770 2.6204 2.7167 2.7969

1.3840 2.1332 2.5220 2.7630 2.9187 3.0223 3.1121

1.4048 2.0327 2.3520 2.5376 2.6578 2.7502 2.8245

Table 4 The average RMS values of priority vectors under diﬀerent scales and prioritizations Size of priority

Saaty scale

Geometrical scale ðc ¼ 2Þ

Ma-Zheng scale

Salo scale ðe ¼ 1=20Þ

Salo scale ðe ¼ 1=17Þ

EVM

LLSM

EVM

LLSM

EVM

LLSM

EVM

LLSM

EVM

LLSM

2 3 4 5 6 7 8 9

0.0322 0.0269 0.0207 0.0164 0.0134 0.0114 0.0098 0.0085

0.0320 0.0268 0.0208 0.0166 0.0135 0.0113 0.0097 0.0085

0.0182 0.0151 0.0117 0.0092 0.0075 0.0062 0.0053 0.0046

0.0182 0.0151 0.0118 0.0092 0.0075 0.0062 0.0052 0.0045

0.0162 0.0154 0.0124 0.0100 0.0083 0.0070 0.0060 0.0052

0.0164 0.0154 0.0124 0.0099 0.0081 0.0069 0.0059 0.0050

0.0169 0.0148 0.0120 0.0093 0.0078 0.0067 0.0058 0.0050

0.0163 0.0147 0.0117 0.0094 0.0078 0.0066 0.0056 0.0049

0.0148 0.0133 0.0108 0.0088 0.0072 0.0060 0.0052 0.0044

0.0147 0.0133 0.0107 0.0086 0.0070 0.0058 0.0049 0.0042

we also ﬁnd that this empirical evidence shows that the geometrical scale ðc ¼ 2Þ is better than Saaty’s scale. (3) The Ma-Zheng scale andthe Salo-Ha¨ma¨la¨inen scale e ¼ 201 and e ¼ 171 . Many researchers [20,37], including ourselves, show that these two scales are better than the Saaty scale. However, from Table 5, we ﬁnd that these two scales are not supported by Saaty’s empirical evidence. Saaty’s empirical evidence shows that the Saaty scales is better than these two scales.

Based on the above analysis, we show that it is more reasonable to set the geometrical scale ðc ¼ 2Þ as the scale function in AHP. In our future research, more simulation and analysis will be done to investigate the inﬂuence of diﬀerent scales and prioritization methods on the AHP decision making process. Moreover, many researchers focus on comparing EVM and LLSM. There is some dispute about which method is better. LLSM is thought to be better in [2,5,32,41], while the Saaty school

Table 5 The RMS values of Saaty’s examples under diﬀerent scales and prioritizations when using Saaty’s method Scale

Prioritization

Example 1

Example 2 (First Trail)

Example 2 (Second Trail)

Example 3

Example 4

Average values of these examples

Saaty scale

EVM LLSM

0.0453 0.0479

0.0289 0.0289

0.0180 0.0160

0.0366 0.0378

0.0298 0.0321

0.0317 0.0325

Geometrical scale ðc ¼ 2Þ

EVM LLSM

0.0346 0.0365

0.0211 0.0213

0.0149 0.0149

0.0335 0.0346

0.0190 0.0230

0.0246 0.0261

Ma-Zheng scale

EVM LLSM

0.0863 0.0861

0.2132 0.2136

0.2270 0.2273

0.1606 0.1656

0.1224 0.1240

0.1619 0.1633

Salo scale ðc ¼ 1=20Þ

EVM LLSM

0.0642 0.0635

0.1423 0.1439

0.1555 0.1556

0.1206 0.1212

0.0842 0.0852

0.1134 0.1139

Salo scale ðc ¼ 1=17Þ

EVM LLSM

0.1253 0.0949

0.0784 0.0802

0.0893 0.0897

0.0525 0.0583

0.0385 0.0436

0.0768 0.0733

Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

[18,25,28,29] holds an opposite opinion. Herman and Koczkodaj [13] show that the diﬀerence between EVM and LLSM is very small. The results (Tables 1–5), which are obtained by Algorithms I,II and Saaty’s method, also show that the eﬀects of these two prioritization methods are similar when the same scale function is adopted. It’s worth mentioning that Kumar and Ganesh [18] show that EVM is better than LLSM using the simulation method presented by Triantaphyllou and Mann [34] (similar to Algorithm II), which is the evidence to claim that EVM is superior to LLSM. But when we use Algorithm II which overcomes the ﬂaw in the simulation method in Triantaphyllou and Mann [34], we obtain diﬀerent conclusions (see Table 4).

5. Discussion 5.1. The parameter of geometrical scale In the above section, we suggest to set the geometrical scale as the scale function in AHP. But when using the geometrical scale, we will get a very diﬀerent priority vector if the parameter (c) takes diﬀerent values [17]. So it’s important to determine the parameter (c) in order to derive more valid priority vectors. Saaty [27] also points out that the difﬁculty to use the geometrical scale is how to determine the scale parameter. It seems to be natural that we can use Algorithm I to search for an optimal parameter copt by minimizing dðD; DÞ. But, in Section 4.1 we ﬁnd that the average values of dðD; DÞ have no obvious diﬀerence when the parameter c equals 1.5, 2 or 4 (see Table 1). At the same time, if we adopt the geometrical scale as the scale function, and use LLSM, not EVM, as the prioritization method in Algorithm I, we have vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uX n X n n 1u 1X dðD; DÞ ¼ t ðIðd ij Þ ðIðd ik Þ Iðd jk ÞÞÞ2 : n i¼1 j¼1 n k¼1 It is shown that the values of dðD; DÞ are independent of the parameter c. Thus, it is impossible to ﬁnd copt by Algorithm I. Next, we use Algorithm II to search for copt. For each diﬀerent priority size (n), we generate 10,000 random priority vectors. We obtain the average values of RMS under diﬀerent parameter values of the

239

geometrical scale (c). The results are shown in Table 6. Table 6 shows that copt is close to 2, which is consistent with Ji and Jiang [17]. Ji and Jiang [17] use Saaty’s four examples to ﬁnd the optimal parameter copt. By minimizing RMS, they also show that copt is close to 2. 5.2. A new prioritization method In this subsection, we design a prioritization method by minimizing dðD; DÞ. Assume that the selected scale function is f ðxÞ. Let F ðwÞ ¼ dðD; DÞ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ﬃ 1 Xn Xn wi ¼ d ij f 1 : i¼1 j¼1 n wj

ð12Þ

A reasonable priority vector w* should be determined so as to minimize F ðwÞ, that is, 8 F ðw Þ ¼ minðF ðwÞÞ > < w ð13Þ n > s:t: wi P 0; P wi ¼ 1: : i¼1

This prioritization method has a deﬁnite physical implication. It makes the semantics of the priority vector close to the semantics of the LPCM given by a decision maker to the greatest degree. When the geometrical scale is used as a scale function, we obtain Theorem 3. Theorem 3. The LLSM is a prioritization method that minimizes dðD; DÞ, if the geometrical scale is used as the scale function. Proof. Since the geometrical scale is adopted as the scale function, we have x

f ðxÞ ¼ c2 :

ð14Þ

Denote A ¼ ðaij Þnn as the numerical pairwise comparison matrix, then d ij

aij ¼ c 2 :

ð15Þ

From (12) and (15), it can be obtained that Xn Xn 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F ðwÞ ¼ ðd ij logpﬃc ðwi Þ þ logpﬃc ðwj ÞÞ2 : i¼1 j¼1 n ð16Þ From (15), we have d ij ¼ logpcﬃ ðaij Þ:

ð17Þ

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Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

Table 6 The average RMS values for diﬀerent parameter values of the geometrical scale c

n

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

2

3

4

5

6

7

8

9

0.1162 0.0709 0.0445 0.0325 0.0250 0.0203 0.0188 0.0178 0.0180 0.0184 0.0189 0.0197 0.0204 0.0211 0.0223 0.0230 0.0239 0.0248 0.0254 0.0263

0.1010 0.0626 0.0399 0.0276 0.0213 0.0179 0.0158 0.0152 0.0150 0.0150 0.0154 0.0159 0.0165 0.0171 0.0175 0.0182 0.0186 0.0193 0.0198 0.0206

0.0821 0.0515 0.0332 0.0233 0.0172 0.0143 0.0129 0.0120 0.0117 0.0118 0.0120 0.0122 0.0127 0.0130 0.0134 0.0138 0.0143 0.0147 0.0152 0.0154

0.0687 0.0432 0.0277 0.0190 0.0142 0.0115 0.0102 0.0095 0.0093 0.0091 0.0094 0.0096 0.0098 0.0103 0.0104 0.0108 0.0111 0.0114 0.0118 0.0121

0.0587 0.0369 0.0236 0.0163 0.0119 0.0097 0.0084 0.0078 0.0075 0.0075 0.0075 0.0077 0.0079 0.0082 0.0084 0.0087 0.0089 0.0093 0.0095 0.0097

0.0506 0.0322 0.0204 0.0141 0.0102 0.0083 0.0072 0.0065 0.0062 0.0062 0.0063 0.0064 0.0065 0.0067 0.0069 0.0071 0.0073 0.0076 0.0077 0.0080

0.0446 0.0282 0.0185 0.0124 0.0091 0.0072 0.0062 0.0055 0.0053 0.0053 0.0053 0.0054 0.0055 0.0057 0.0058 0.0060 0.0062 0.0064 0.0065 0.0067

0.0401 0.0251 0.0165 0.0111 0.0080 0.0063 0.0054 0.0048 0.0046 0.0045 0.0045 0.0046 0.0047 0.0048 0.0050 0.0052 0.0053 0.0055 0.0056 0.0058

From (16) and (17), we can get vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uX n X n 1u F ðwÞ ¼ t ðlogpcﬃ ðaij Þ logpcﬃ ðwi Þ þ logpcﬃ ðwj ÞÞ2 : n i¼1 j¼1 ð18Þ

m LPCMs, Dk ¼ ðd kij Þnk nk ðk ¼ 1; 2; . . . ; mÞ. Denote Dk ¼ ðd kij Þnk nk ðk ¼ 1; 2; . . . ; mÞ as the LPCMs of the derived priority vectors. If we derive the priority vectors by LLSM, then we have d kij ¼ ﬃ1 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Q nk nk ðf ðd kip ÞÞ p¼1 A. By minimizing J ðf ðxÞÞ ¼ f 1 @ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Q nk

nk

Let

Pm

i¼1

j¼1

þ logpcﬃ ðwj ÞÞ : 2

ð19Þ

If w* is the priority vector derived by LLSM, we have Gðw Þ ¼ minw ðGðwÞÞ. Thus F ðw Þ ¼ minðF ðwÞÞ:

ð20Þ

w

This complete the proof of Theorem 3.

h

5.3. An optimization model to select an appropriate numerical scale for the AHP decision makers In the above sections, we compare several numerical scales in the average sense. But diﬀerent decision makers may use diﬀerent numerical scales. Therefore, it is important to study how to ﬁnd the appropriate numerical scales for diﬀerent decision makers. Assume that an AHP decision maker gives

ðf ðd kjp ÞÞ

; Dk Þ, we can ﬁnd an optimal scale function f ðxÞ for the decision maker. In terms of the definition of a scale function (Deﬁnition 3), we know that f ðxÞ needs to satisfy the following properties: k¼1 dðD

n X n X GðwÞ ¼ ðlogpcﬃ ðaij Þ logpﬃc ðwi Þ

p¼1

k

(1) f ðxÞ > 0; (2) f ðxÞ is a monotonically increasing function, namely f 0 ðxÞ > 0; (3) f ðxÞ f ðxÞ ¼ 1. As a result, we construct the following optimization model. vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 0 0 0 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 112ﬃ1 u nk > Q u > nk k > f ðd Þ > BP u nk P nk B > B p¼1 ip CC C u 1 P > B > Bd k f 1 B rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ CC ﬃC > min B mk¼1 u > ij n @ n Qk AA C tnk ðnk 1Þ i¼1 j¼1 @ > A < f @ k f ðd k Þ jp

> > > f ðxÞ > 0 > > > > 0 > f ðxÞ > 0 > > : f ðxÞ f ðxÞ ¼ 1:

p¼1

Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

Analytically, this model is diﬃcult to solve. However, it has its practical signiﬁcance. Let f be the set of the existing scale functions (the Saaty scale, the Ma-Zheng scale, the geometrical scale and the Salo-Ha¨ma¨la¨inen scale, etc). We can easily ﬁnd a relatively appropriate scale function f ðxÞ for the decision maker, that is J ðf ðxÞÞ 0

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 0 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ112 1 u u nk Q u B B B nk f ðd kip ÞCC C u BX nk X nk B B p¼1 CC C X B m u 1 B k B CC C u 1 B ¼ min B : Bd ij f B sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃCC C u n f 2f B B B CC C n ðn 1Þ u k k k i¼1 j¼1 @ @ nk Q f ðd k ÞAA C @ k¼1 t A jp

p¼1

It is clear that the time complexity of ﬁnding f ðxÞ from f is Oðn3 Þ. 6. Conclusions In this paper, we present two performance measure algorithms, and carry out a detailed comparative study on the numerical scales and the prioritization methods. The major contributions and ﬁndings are as follows: (1) By using our two performance measure algorithms, we compare several common numerical scales (the Saaty scale, the Ma-Zheng scale, the geometrical scale and the SaloHa¨ma¨la¨inen scale). Synthesizing other researchers’ comparative results, we show that it is more reasonable to set the geometrical scale ðc ¼ 2Þ as the scale function in AHP. (2) We also compare two prioritization methods (EVM and LLSM). We show that the eﬀects of these two prioritization methods are similar. As an additional conclusion, Kumar and Ganesh [18] show that EVM is better than LLSM using the simulation method of Triantaphyllou and Mann [34]. However, when we use Algorithm II, which overcomes the ﬂaw in Triantaphyllou and Mann’s method, we obtain diﬀerent conclusions. (3) We adopt Algorithm II to determine the parameter of the geometrical scale. We ﬁnd that the optimal parameter of geometrical scale is close to 2. (4) If the geometrical scale is used as the scale function, the paper proves that LLSM is the prioritization method that can minimize the diﬀerence between the initial LPCM given by

241

a decision maker and the LPCM of the corresponding priority vector. (5) We construct an optimization model to select appropriate numerical scales for diﬀerent decision makers. Acknowledgements We are very grateful to the editor and the anonymous referees for their valuable comments and suggestions. This research is supported by National Natural Science Foundation of China (NSFC) (No. 70121001, 70471035 and 70525004) and the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CUHK4443/ 04H). References [1] K.J. Arrow, Social Choice and Individual Values, second ed., Wiley, New York, 1963. [2] J. Barzilai, Deriving weights from pairwise comparison matrices, Journal of the Operational Research Society 48 (1997) 1226–1232. [3] V. Belton, T. Gear, On a shortcoming of saaty’s method of analytic hierarchies, Omega 11 (3) (1983) 228–230. [4] E.U. Choo, W.C. Wedley, A common framework for deriving preference values from pairwise comparison matrices, Computers & Operations Research 31 (2004) 893–908. [5] G. Crawford, C. Williams, A note on the analysis of subjective judgement matrices, Journal of Mathematical Psychology 29 (1985) 387–405. [6] Y.C. Dong, Y.F. Xu, Consistency measures of linguistic preference relations and its properties in group decision makingLecture Notes in Artiﬁcial Intelligence, vol. 4223, Springer-Verlag, Germany, 2006, pp. 501–511. [7] J.S. Dyer, Remarks on the analytic hierarchy process, Management Science 36 (3) (1990) 249–258. [8] J.S. Dyer, A clariﬁcation of remarks on the analytic hierarchy process, Management Science 36 (3) (1990) 274– 275. [9] Z.P. Fan, Y.P. Jiang, A judgment method for the satisfying consistency of linguistic judgment matrix, Control and Decision 19 (2004) 903–906. [10] J.S. Finan, W.J. Hurley, Transitive calibration of the AHP verbal scale, European Journal of Operational Research 112 (1999) 367–372. [11] E.H. Forman, S.L. Gass, The analytic hierarchy process—an exposition, Operations Research 49 (2001) 469–486. [12] P.T. Harker, L.G. Vargas, The theory of ratio scale estimation: Saaty’s analytic process, Management Science 33 (1987) 1383–1403. [13] M.W. Herman, W.W. Koczkodaj, A monte carlo study of pairwise comparison, Information Processing Letters 57 (1996) 25–29. [14] F. Herrera, A sequential selection process in group decision making with linguistic assessment, Information Sciences 85 (1995) 223–239.

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Y. Dong et al. / European Journal of Operational Research 186 (2008) 229–242

[15] F. Herrera, E. Herrera-Viedma, Choice functions and mechanisms for linguistic preference relations, European Journal of Operational Research 120 (2000) 144–161. [16] E. Herrera-Viedma, F. Herrera, F. Chiclana, M. Luque, Some issues on consistency of fuzzy preference relations, European Journal of Operational Research 154 (2004) 98– 109. [17] P. Ji, R. Jiang, Scale transitivity in the AHP, Journal of the Operational Research Society 54 (2003) 896–905. [18] V.N. Kumar, L. Ganesh, A simulation-based evaluation of the approximate and the exact eigenvector methods employed in AHP, European Journal of Operational Research 95 (1996) 656–662. [19] F.A. Lootsma, Scale sensitivity in the multiplicative AHP and SMART, Journal of Multi-Criteria Decision Analysis 2 (1993) 87–110. [20] D. Ma, X. Zheng, 9/9-9/1 Scale method of AHP, in: Proceedings of the Second International Symposium on the AHP, University of Pittsburgh, Pittsburgh, PA, 1991, pp. 197–202. [21] J. Pe´rez, Some comments on Saaty’s AHP, Management Science 41 (1995) 1091–1095. [22] T.L. Saaty, A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 15 (1977) 234–281. [23] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [24] T.L. Saaty, An exposition of the AHP in reply to the paper ‘‘remarks on the analytic hierarchy process’’, Management Science 36 (3) (1990) 259–268. [25] T.L. Saaty, Eigenvector and logarithmic least squares, European Journal of Operational Research 48 (1990) 156– 160. [26] T.L. Saaty, Fundamentals of Decision Making, RSW Publications, 1994. [27] T.L. Saaty, Highlights and critical points in the theory and application of the analytic hierarchy process, European Journal of Operational Research 74 (1994) 426– 447.

[28] T.L. Saaty, G. Hu, Ranking by the eigenvector versus other methods in the analytic hierarchy process, Applied Mathematical Letters 11 (4) (1998) 121–125. [29] T.L. Saaty, Decision-making with the AHP: Why is the principal eigenvector necessary, European Journal of Operational Research 145 (2003) 85–91. [30] A.A. Salo, R.P. Ha¨ma¨la¨inen, On the measurement of preferences in the analytic hierarchy process, Journal of Multi-Criteria Decision Analysis 6 (1997) 309–319. [31] B. Srdjevic, Combining diﬀerent prioritization methods in the analytic hierarchy process synthesis, Computers & Operations Research 32 (2005) 1897–1919. [32] E. Takeda, K. Cogger, P.L. Yu, Estimating criterion weights using eigenvectors: A comparative study, European Journal of Operational Research 29 (1987) 360–369. [33] L.L. Thurstone, A law of comparative judgements, Psychological Review 34 (1927) 273–286. [34] E. Triantaphyllou, S.H. Mann, An evaluation of the eigenvalue approach for determining the membership values in fuzzy sets, Fuzzy Sets and Systems 35 (1990) 295–301. [35] O.S. Vaidya, S. Kumar, Analytic hierarchy process: An overview of applications, European Journal of Operational Research 169 (2006) 1–29. [36] L. Vargas, An overview of analytic hierarchy process: Its applications, European Journal of Operational Research 48 (1) (1990) 2–8. [37] Z.S. Xu, A simulation-based evaluation of several scales in the analytic hierarchy process, Systems Engineering— Theory & Practice 20 (7) (2000) 58–62. [38] Z.S. Xu, Uncertain Attribute Decision Making: Method and Applications, Tsinghua University Press, Beijing, 2004. [39] Z.S. Xu, Deviation measures of linguistic preference relations in group decision making, Omega 33 (2005) 249–254. [40] S. Yuuji, An evaluation of judgment scale in the analytic hierarchy process, Journal of the Operations Research Society of Japan 42 (1) (1999) 77–82. [41] F. Zahedi, A simulation study of estimation methods in the analytic hierarchy process, Socio-Economic Planning Science 20 (1986) 347–354.

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