A method to calculate the ranges of criteria weights in ELECTRE I and II methods

Computers & Industrial Engineering 137 (2019) 106067

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A method to calculate the ranges of criteria weights in ELECTRE I and II methods Xianliang Liu, Shu-ping Wan

T

⁎

College of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China

ARTICLE INFO

ABSTRACT

Keywords: Decision analysis Multiple criteria decision-making ELECTRE I ELECTRE II Sensitivity analysis

In this study, the ranges of criteria weights based on the outranking relations in ELECTRE I and II methods are analyzed, which is considered as the sensitivity analysis regarding criteria weights. First, an important theorem is proved, in which the maximum value of discordance indices for any two alternatives is shown to be equal to 1. Using this theorem, four situations of outranking relations for any two alternatives are simplified to two situations when sensitivity analysis of criteria weights is performed for ELECTRE I. Subsequently, some inequalities are obtained to calculate the ranges of criteria weights when keeping the outranking relations between any two alternatives unchanged. For ELECTRE II, nine situations of strong and weak outranking relations are simplified into three situations of relations according to this theorem. The ranges of criteria weights are also derived when the strong and weak outranking relations remain unchanged. Finally, two examples are presented to test and verify the effectiveness of the sensitivity analysis of criteria weights for ELECTRE I. Two analogous examples are also presented in ELECTRE II.

1. Introduction Multiple criteria decision-making (MCDM), as the most well-known branch of decision-making, has been widely used in many fields, such as the big data (Ullaha & Noor-E-Alam, 2018), supporting performance appraisal (Marijana, Nataša, Mladen, & Ivan, 2018), facility layout design selection (Maniya & Bhatt, 2011). The aim of MCDM problems is to rank a finite number of alternatives rationally, where each alternative is explicitly described in terms of multiple criteria (also called attributes). Typically, MCDM can be easily expressed by an m × n decision matrix (see Table 1 below), where the element gk (ai ) in the decision matrix indicates the performance value of alternative ai on criteria ck (i = 1, 2, , m;k = 1, 2, , n ). It is often assumed that the decision maker determines the weight of criteria ck , denoted as wk , where wk (0, 1) and 1 k n wk = 1. Many methods have been proposed for MCDM, such as the elimination et choice translating reality (ELECTRE), technique for order preference by similarity to an ideal solution (TOPSIS) (Hwang & Yoon, 1981), analytic hierarchy process (AHP) (Saaty, 1980), and measuring attractiveness by a categorical-based evaluation technique (MACBETH) (Costa, DeCorte, & Vansnick, 2005). Until now, there are many researches regarding MCDM. A new method was introduced in Yu, Zhang, Zhong, and Sun (2017) to deal with multi-criteria group decision

⁎

making problems with unbalanced hesitant fuzzy linguistic term sets. Using the logarithmic least squares method, Zhang, Kou, Yu, and Guo (2018) proposed an approach to deriving a priority weight vector from an incomplete hesitant fuzzy preference relation. Based on a novel computational model, an approach to linguistic large-scale multiattribute group decision making was presented in Zhang, Guo, and Martínez (2017). Zhang and Guo (2017) gave some studies about group decision making with the intuitionistic multiplicative preference relations. ELECTRE is regarded as one of the popular MCDM methods and has been widely used in real life, such as cloud services (Ma, Zhu, Hu, Li, & Tang, 2017), renewable energy policy selection (Mousavi, Gitinavard, & Mousavi, 2017), and small hydropower plant investments (Saracoglu, 2015). The primary idea of ELECTRE is the utilization called “outranking relations.” ELECTRE has evolved into a number of other variants, such as ELECTRE IV, ELECTRE IS, ELECTRE TRI, ELECTRE TRI-C, and ELECTRE TRI-NC (Almeida-Dias, Figueira, & Roy, 2010, 2012; Greco, Figueira, & Ehrgott, 2005). Currently, the most widely used versions are ELECTRE I, ELECTRE II and ELECTRE III (Greco et al., 2005). The advantage of ELECTRE is that the imperfect knowledge is modeled by considering the indifference and preference thresholds. Nevertheless, ELECTRE cannot deal with purely ordinal scales, for the

Corresponding author. E-mail address: [email protected] (S.-p. Wan).

https://doi.org/10.1016/j.cie.2019.106067 Received 7 March 2019; Received in revised form 7 September 2019; Accepted 14 September 2019 Available online 19 September 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.

Computers & Industrial Engineering 137 (2019) 106067

X. Liu and S.-p. Wan

(iii) Vetschera (1986) performed the sensitivity analysis of ELECTRE I from the standpoint of theoretics, where the parameters were not constrained. Although there are three parameters considered, namely the concordance threshold, the discordance threshold and the weight, it only considered how to obtain the ranges for these parameters to keep E-set unchanged, where the E-set is a set of the best alternatives. Moreover, all analyses were carried out for only one criteria weight under the assumption that the weights for all other criteria remain unchanged.

Table 1 Structure of decision matrix. c1

c2

cn

a1

g1 (a1)

g2 (a1)

gn (a1)

a2

g1 (a2)

g2 (a2)

gn (a2)

am

g1 (am)

g2 (am)

gn (am)

discordance index, a metric scale is required since differences are formed and compared. For example, Figueira, Mousseau, and Roy (2005) explicitly stated about ELECTRE I that it should be applied only when all the criteria have been coded in numerical scales. In addition, Wang and Triantaphyllou (2008) found that the rank reversal phenomenon occurs in ELECTRE II and III. That is to say, the relative rankings of two alternatives could be reversed when an alternative is added or deleted. A new MCDM method based on proximity indexed value for minimizing rank reversals was presented in Mufazzal and Muzakkir (2018). The sensitivity and robustness analyses of MCDM methods have attracted increasing attention in recent years. Considering the criteria weights, Genc (2014) performed the sensitivity analyses for the preference ranking organization method for enrichment evaluations (PROMETHEE) and TOPSIS by an example of car quality assessment. Wan, Wang, and Dong (2016) showed the sensitivity analyses for the quantifier guided non-dominance degree (QGNDD) of alternatives for different tolerance parameters in different approaches. Under the hypothesis that all the original parameter weights of TOPSIS are equal, Li, Qian, Wu, and Chen (2013) showed that how the assessment results would be affected in water quality assessment when each weight was changed separately. Under another hypothesis that all indices of a given sample were changed simultaneously with the same ratio, Li et al. (2013) performed the sensitivity analysis of TOPSIS in water quality assessment. Based on a flood damage case, a TOPSIS-based robustness uncertainty sensitivity (RUS) approach was conducted in Song and Chung (2016). The strategic weight manipulation was studied in Dong, Liu, Liang, Chiclana, and Herrera-Viedma (2018), Liu, Dong, Liang, Chiclana, and Herrera-Viedma (in press-b); Liu, Zhang, Wu, and Dong, in press-a. In Dong et al. (2018), mixed 0–1 linear programming models were proposed to obtain the ranking range of an alternative. These models are used to provide the best and worst ranking of the alternative. Additionally, some conditions were presented to manipulate a strategic attribute weight when the decision maker wants to obtain her/ his desired ranking based on the ranking range. In Liu et al. (in press-b), the strategic weight manipulation with minimum cost was studied to obtain a desired ranking of alternatives in a group decision context with interval criteria weights information. In ELECTRE I, sensitivity analyses for parameters such as concordance threshold, discordance threshold and weight were first studied in Vetschera (1986). To the best of our knowledge, only Vetschera (1986) studied the sensitivity analysis for ELECTRE I. The aforementioned studies have significantly enriched the sensitivity analyses of MCDM methods. However, the limitations are as follows.

From the above mentioned problems or phenomena, the sensitivity and robustness analyses of MCDM methods are necessary, and are important. That is the purpose and significance of this paper. We conduct sensitivity analyses of classical ELECTRE I and II. The focus is to obtain the ranges for criteria weights while keeping the outranking relations between two arbitrary alternatives unchanged. Because the number of outranking relations between alternatives is large, the challenge in this study is to maintain all the outranking relations when some criteria weights are changed. The main characteristics of this study is to obtain the ranges of some criteria weights by constructing linear inequalities regarding those criteria weights. Meanwhile, the number of outranking relations which need to be considered between alternatives is reduced by at least half using the properties of outranking relations. The primary contributions of this paper are summarized as follows: (1) For the discordance indices of any two alternatives in ELECTRE I and II, an important theorem is proven, i.e., the maximum value of discordance indices for pairs (ai , aj ) and (aj , ai ) is equal to 1. (2) Some desirable properties of outranking relations for ELECTRE I, and strong and weak outranking relations for ELECTRE II are discussed in detail. (3) For ELECTRE I, keeping four situations (ai aj , aj ai , ai aj and ai aj ) of outranking relations for any two alternatives ai and aj unchanged is transformed into keeping two situations ( when D (ai , aj ) ) unchanged, when the sensitivity analysis of criteria weights is performed. Subsequently, some inequalities are obtained to calculate the ranges of criteria weights when keeping the outranking relations between any two alternatives unchanged. (4) For ELECTRE II, nine types of strong and weak outranking relations (see below) are simplified into three relations when the sensitivity analysis of criteria weights is performed. Subsequently, some inequalities are acquired to compute the ranges of criteria weights when the outranking relations between any two alternatives are unchanged. The remainder of this paper is organized as follows. Section 2 reviews ELECTRE I and analyzes the sensitivity of criteria weights for ELECTRE I. Similarly, ELECTRE II is introduced and some results of the sensitivity analysis of ELECTRE II are proposed in Section 5. Subsequently, Section 3, 4 and Section 6, 7 present four examples to verify the effectiveness of the ranges of criteria weights in ELECTRE I and II, respectively. The conclusions are presented in the last section. 2. Sensitivity analysis of ELECTRE I In this section, ELECTRE I is reviewed and its sensitivity analysis is presented. Suppose that the alternative set A = {a1, a2 , , am} , the criteria set C = {c1, c2, , cn} and J = {1, 2, , n} . Assume that gk (ai ) expresses the performance value of alternative ai on criterion ck . Let J > (ai , aj ) = {k|gk (ai) > gk (aj )} j ), (i , j = 1, 2, , n, i where J > (ai , aj ) represents the subscript set of criteria that satisfies the performance value of alternative ai being strictly greater than aj . Similarly, J = (ai , aj ) = {k|gk (ai ) = gk (aj )} indicates the subscript set of criteria that satisfies the performance value of alternative ai being equal to aj . J < (ai , aj ) = {k|gk (ai) < gk (aj )} expresses the subscript set of criteria that satisfies the performance value of alternative ai being strictly less

(i) Hitherto, the study of sensitivity analyses of MCDM has been rare. Furthermore, most of sensitivity analysis methods for MCDM were performed using only some numerical examples, such as (Genc, 2014; Song & Chung, 2016; Wan et al., 2016). They lacked theoretical analysis. Therefore, the flexibility of these methods was low. (ii) Although literatures (Li et al., 2013; Li, Qian, Wu, & Chen, 2013) performed sensitivity analyses from the theoretical point of view, they imposed some restrictions on the parameters of MCDM methods. 2

Computers & Industrial Engineering 137 (2019) 106067

X. Liu and S.-p. Wan

than aj . Moreover, let J (ai , aj ) = J > (ai , aj ) J = (ai , aj ) and J (ai , aj ) = J < (ai , aj ) J = (ai , aj ) . There are two primary stages for most ELECTRE methods. The first stage is the construction of the outranking relations of alternatives. The second stage is to obtain the final ranking of alternatives exploiting these outranking relations. Different ELECTRE methods have different definitions of the outranking relations of alternatives and diverse exploitations of these relations to obtain the final ranking for alternatives. Moreover, ELECTRE can build one or several(crispy, fuzzy or embedded) outranking relations. ELECTRE I builds one outranking relation . The outranking relation ai aj implies that the alternative ai is at least as good as aj . Considering two arbitrary alternatives ai and aj , four situations may occur as follows:

(aj , ai ) can be seen as two different pairs. Considering the outranking relation of the pair (ai , aj ) , two situations may occur: ai aj or not ai aj (denoted by ). Similarly, for the pair (aj , ai ) , there also exist two situations: aj ai or . Hence, for any two alternatives ai and aj , they may have four situations: ai aj and aj ai , that is ai aj;ai aj and , that is and aj ai , that is and , that is ai aj . The possible outranking relations for any two alternatives ai and aj are sophisticated. In fact, some situations can be simplified by the following conclusions. Theorem 2.1. For any two alternatives ai and aj , the discordance indices between them satisfy the following equality

max{D (ai , aj ), D (aj , ai )} = 1. Proof. By comparing the performance values of alternatives ai and aj , the following cases must be discussed.

• a a and not a a , which implies that a is strictly preferred to a , denoted by a a ; • a a and not a a , which implies that a is strictly preferred to a , denoted by a a ; • a a and a a , which implies that a is indifferent to a , denoted by a a; • Not a a and not a a , which implies that a is incomparable to a , i

j

j

i

j

i

i

j

i

j

i

j

i

i

j

j

j

i

(1) If all the performance values of ai are greater than or equal to those of aj , then

j

i

i

max{max{gk (aj )

D (a i , a j )=

j

i

j

i

j

denoted by ai aj .

j

i

i

The construction of an outranking relation concepts:

C (ai , aj ) =

D (ai , aj ) =

=

max{max{gk (ai)

D (aj , ai)=

gk (ai )}, 0}

max{|gk (ai ) k J

gk (aj )|}

.

and C (ai , aj )

max{gk (ai)

gk (aj )}

k (ai )

gk (aj )}

=max{g k J

= 1.

(2) If all the performance values of aj are greater than or equal to those of ai , then max{max{gk (ai)

D (aj , ai)=

gk (aj )}, 0}

k J

max{| gk (ai)

gk (aj ) |}

k J

0

=max{g

k (aj )

k J

=0

gk (ai)}

and max{max{gk (aj )

D (a i , a j )=

gk (ai)}, 0}

k J

max{| gk (ai)

gk (aj ) |}

k J

max{gk (aj )

gk (ai)}

k (aj )

gk (ai)}

k J

=max{g k J

= 1.

gk (ai ) > gk (aj ) k = 1, 2, , t , gk (ai ) = gk (aj ) (3) If for k = t + 1, t + 2, , t + l gk (ai ) < gk (aj ) and k = t + l + 1, t + l + 2, , n , then without loss of generality,

.

After computing the concordance and discordance indices for all pairs of alternatives, all the outranking relations are built by comparing [0.5, 1) and these indices with the concordance threshold value (0, 1) , where discordance threshold value and are the parameters provided in advance. For the pair of alternatives (ai , aj ) , the outranking relation ai aj is built by the following two rules: (1) C (ai , aj ) (2) D (ai , aj )

gk (aj ) |}

k J

k J (ai, aj )

k J

gk (aj )}, 0}

k J

max{| gk (ai) k J

wk ,

max{max{gk (aj )

=0

gk (aj )}

and

is based on two major

It is noteworthy that the outranking relation ai aj holds if and only if the concordance and discordance conditions are fulfilled. In ELECTRE I, two indices, namely concordance index and discordance index, are used to define the outranking relations. The concordance index C (ai , aj ) measures the strength of the hypothesis that alternative ai is at least as good as aj , the discordance index D (ai , aj ) measures the strength of evidence against such hypothesis. The concordance index and the discordance index for each pair of alternatives (ai , aj ) are defined, respectively, as follows:

wk

0

k (ai )

k J

1. Concordance: the outranking relation ai aj is to be validated if a sufficient majority of criteria supports this assertion; 2. Discordance: as a prerequisite for the concordance condition to hold, none of the criteria in the minority should oppose too strongly to the assertion ai aj .

k J (ai, aj ) n w k=1 k

gk (aj ) |}

k J

=max{g

j

gk (ai)}, 0}

k J

max{| gk (ai)

g1 (ai )

g1 (aj ) = max{|gk (ai ) k J

gk (aj )|}.

Thus, we have max{max{gk (aj )

D (a i , a j )=

gk (aj ) |}

k J

max

C (aj , ai ) ;

gk (ai)}, 0}

k J

max{| gk (ai)

t +l +1 k n

=

{gk (aj )

g1 (ai)

gk (ai)}

g1 (aj )

and

The outranking relation ai aj holds if and only if the two rules are satisfied simultaneously. In this following, the outranking relations are analyzed. Subsequently, the ranges of criteria weights are studied when the outranking relations of any two alternatives remain unchanged. Because concordance index C (ai , aj ) may be different from C (aj , ai ) , discordance index D (ai , aj ) is different from D (aj , ai ) , the pair (ai , aj ) and

max{max{gk (ai)

D (aj , ai)=

k J

max{| gk (ai) k J

g (ai)

=g1 (a ) 1

3

i

g1 (aj ) g1 (aj )

gk (aj )}, 0} gk (aj ) |}

= 1.

1

for for

Computers & Industrial Engineering 137 (2019) 106067

X. Liu and S.-p. Wan

To summarize, we find that D (ai , aj ) [0, 1] and D (aj , ai ) [0, 1]. Moreover, at least one of D (ai , aj ) and D (aj , ai ) is equal to 1. Hence, the discordance indices between ai and aj satisfy

keeping the following two situations unchanged. (1) ai aj ; (2) when D (ai , aj )

max{D (ai , aj ), D (aj , ai )} = 1.

Without loss of generality, suppose that the former r weights w1, w2, , wr are changed to w1 + w1, w2 + w2, , wr + wr , respectively. It is noteworthy that 2 r n because the sum of all the weights is equal to 1. Obviously, all the variables w1, w2, , wr must satisfy w1 + w2 + + wr = 0 and wk ( wk , 1 wk ) for k = 1, 2, , r . In the following, sensitivity analysis of criteria weights for ELECTRE I is discussed. The aim is to obtain the ranges of w1, w2, , wr for keeping the outranking relation ai aj or (when D (ai , aj ) ) unchanged for any two alternatives ai and aj j ). (i For each pair of alternatives (ai , aj ) , it is assumed that Jg {1, 2, , r } J > (ai , aj ), Je {1, 2, , r } J = (ai , aj ) Jl and {1, 2, , r } J < (ai , aj ) . If ai and aj satisfy ai aj , then the following theorem is proposed.

Generally, the performance values for any two alternatives cannot be exactly equal. If two alternatives have the same performance values on each criterion, then the two alternatives can be regarded as the same in decision-making. Thus, the case that gk (ai ) = gk (aj ) for all k = 1, 2, , n is not considered herein. According to the two rules of (0, 1) , the outranking relations and the discordance threshold value following properties hold. Corollary 2.1. For any two alternatives ai and aj , the outranking relations ai aj and aj ai cannot hold simultaneously, i.e., ai aj cannot be established. Proof. According to Theorem 2.1, the threshold

satisfies

< 1 = max{D (ai , aj ), D (aj , ai )}. Thus, the inequalities D (ai , aj ) and D (aj , ai ) simultaneously. This completes the proof. □

Theorem 2.3. The outranking relation ai aj remains unchanged if and only if w1, w2, , wr satisfy

cannot hold

wk

k Jg Je

Corollary 2.2. For two arbitrary alternatives ai and aj , ai aj holds if and only if ai aj holds.

k Jg

k J (ai, aj )

wk

wk

k Jl

wk

k J < (ai, aj )

wk

k J > (ai, aj )

wk

(2-1)

Proof. As D (ai , aj ) does not depend on the criteria weights, D (ai , aj ) always holds. To maintain ai aj , the concordance indices C (ai , aj ) and C (aj , ai ) must satisfy

Proof. Obviously, the necessary part of this corollary is true. For the sufficiency part, if ai aj , then D (ai , aj ) . Owing to max{D (ai , aj ), D (aj , ai )} = 1, the inequality D (aj , ai ) = 1 > holds. Hence, it yields , i.e., ai aj . □

C (a i , a j )

Similarly, aj ai holds if and only if aj ai holds. As ai aj and aj ai can be regarded as the same type of outranking relation, only the outranking relations ai aj and ai aj need to be considered when calculating the ranges of criteria weights under the conditions that all the outranking relations remain unchanged. Corollary 2.3. For each pair of alternatives (ai , aj ) , if D (ai , aj ) ai aj if and only if .

.

C (a i , a j )

C (aj , ai )

(2-2)

The concordance indices C (ai , aj ) and C (aj , ai ) are calculated as follows:

C (a i , a j ) =

, then

C (aj , ai ) =

k J (ai, aj )

wk +

k J (ai, aj )

wk +

k Jg Je

wk

k Je Jl

wk

(2-3)

Proof. The necessary condition is true. For the sufficiency condition, as max{D (ai , aj ), D (aj , ai )} = 1, D (ai , aj ) and we have D (aj , ai ) = 1 > . It is clear that aj ai does not hold. Therefore, ai aj if and only if .□

Combining Inequalities (2-2) with Eqs. (2-3), Inequalities (2-1) can be obtained. □

If D (ai , aj ) > and D (aj , ai ) > , then the outranking relation of ai and aj is definitely ai aj . That is, such a situation does not need to be considered. If D (ai , aj ) , according to Corollary 2.3, only the outranking relation needs to be considered when computing the range of criteria weights under the condition that the outranking relation ai aj remains unchanged.

Theorem 2.4. The outranking relation ai aj does not hold when D (ai , aj ) if and only if w1, w2, , wr satisfy

Theorem 2.2. If the pair of alternatives (ai , aj ) satisfies D (ai , aj ) the outranking relation holds if and only if

1 C (ai , aj ) < ( + C (aj , ai ) + | 2

When the previous outranking relation ai aj does not hold and D (ai , aj ) , a theorem is obtained as follows.

|

wk k J (ai, aj )

+2

, then

wk | > 2 k Je Jl

wk k Jg Je

wk k J (ai, aj )

wk

wk.

k J (ai, aj )

k Je Jl

(2-4)

Proof. As D (ai , aj ) does not depend on the criteria weights, D (ai , aj ) is always unchanged. By Theorem 2.2, such that ai aj still does not hold, the concordance indices C (ai , aj ) and C (aj , ai ) should satisfy

C (aj , ai )|).

Proof. According to the rules of the outranking relation ai aj , this conclusion is obviously holds. □

C (ai , aj ) <

According to Theorem 2.2, the outranking relation ai aj does not D (ai , aj ) > hold if and only if or 1 C (ai , aj ) < 2 ( + C (aj , ai ) + | C (aj , ai )|) . If is unchanged regardless of how the criteria weights change. If D (ai , aj ) , then keeping (that is ai aj in this case) unchanged is C equivalent to maintaining the inequality 1 (ai , aj ) < 2 ( + C (aj , ai ) + | C (aj , ai )|) . Based on the analysis above for the outranking relations, keeping the four situations (ai aj;aj ai;ai aj;ai aj ) of the outranking relations for any two alternatives ai and aj unchanged is transformed into

1 ( + C (aj , ai ) + | 2

C (aj , ai )|)

(2-5)

The concordance indices C (ai , aj ) and C (aj , ai ) are computed as follows:

C (a i , a j ) =

k J (ai, aj )

wk +

k Jg Je

wk

C (aj , ai ) =

k J (ai, aj )

wk +

k Je Jl

wk

(2-6)

Combining Eqs. (2-6) and Inequalities (2-5), Inequalities (2-4) can be derived. By solving Inequalities (2-1) and (2-4) for all pairs of alternatives 4

Computers & Industrial Engineering 137 (2019) 106067

X. Liu and S.-p. Wan

(ai , aj ) that satisfy ai aj or (when D (ai , aj ) ), the ranges of w1, w2, , wr can be obtained. In general, it is easy to solve Inequalities (2-1) and (2-4) when the number of alternatives is not large. If these inequalities are complicated or the number of alternatives is large, the Khachian algorithm (Khachian, 1979) can be used to solve them and can only be derived the values of w1, w2, , wr , but not the ranges.

Table 5 Discordance index.

a1 a2 a3 a4

a1

a2

a3

a4

× 1/3 2/3 1

1 × 1 1

1 0 × 1

1/3 0 0 ×

3. Example 1 of sensitivity analysis of ELECTRE I In this section, an example of sensitivity analysis of ELECTRE I is presented and the ranges are obtained for some criteria weights that are changed. Subsequently, we test and verify that the outranking relation for each pair of alternatives is unchanged when the criteria weights are within the ranges. Finally, the changing of outranking relations is exhibited when the criteria weights are not in the ranges. This example adapted from Wang and Triantaphyllou (2008) is to obtain the best location from four alternatives ai (i = 1, 2, 3, 4) for a waste water treatment plant in Ireland. Seven benefit criteria ck (k = 1, 2, , 7) are considered in this decision. The performance values of alternatives on criteria are given in Table 2. The criteria weights are shown in Table 3. In this example, suppose that the weights of criteria c4 and c5 are changed as 0.35 + w1 and 0.37 + w2 , where w1 and w2 satisfy 0 < 0.35 + w1 < 1, 0 < 0.37 + w2 < 1 and w1 + w2 = 0 .

To obtain the ranges of w1 and w2 while keeping all the outranking relations unchanged, the following steps must be completed. Step 1: Calculate the concordance indices C (ai , aj ) and discordance j . The conindices D (ai , aj ) for i = 1, 2, 3, 4;j = 1, 2, 3, 4 and i cordance and discordance indices of this example are calculated in Tables 4 and 5, respectively. Step 2: Build the outranking relations for each pair of alternatives by comparing these indices with the concordance threshold value (0, 1) . In this ex[0.5, 1) and the discordance threshold value ample, set = 0.6 and = 0.75 . It is noteworthy that the outranking relation of (ai , aj ) cannot be considered when D (ai , aj ) > , because is always true. Hence, only the pairs of alternatives (a2 , a1), (a3 , a1), (a1, a4 ), (a2, a3), (a2, a4), (a3, a4) need to be considered in Table 5. According to the two rules of ELECTRE I in Section 2, the outranking relations can be built as follows:

Step 3: Obtain the ranges of w1 and w2 while keeping the six outranking relations above unchanged. According to Inequalities (2-1) and (24), the six outranking relations above remain unchanged if and only if w1 and w2 simultaneously satisfy the following six groups inequalities.

Table 2 Decision matrix.

a1 a2 a3 a4

c1

c2

c3

c4

c5

c6

c7

1 3 3 1

2 5 5 2

1 3 3 2

5 5 4 5

2 3 3 1

2 3 2 1

4 3 2 1

a2 a1

a3 a1

Table 3 Criteria weights. Criteria Weight

a1 a4

c1

c2

0.07

0.08

c3

c4

0.05

0.35

c5 0.37

c6 0.03

c7

a2 a3

0.05

a2 a4

Table 4 Concordance index.

a1 a2 a3 a4

a3 a4

a1

a2

a3

a4

× 0.95 0.60 0.55

0.40 × 0.57 0.35

0.43 1.00 × 0.35

0.95 1.00 0.65 ×

5

w1 + w 2 w2

k J < (a2, a1)

w2 w2

k J (a2, a1)

k J (a3, a1)

w1

w1 + w 2 w1

w2

wk

wk

k J (a3, a 4 )

w1

k J > (a3, a1)

wk = 0.6

wk

0.55

0.17

0.95 =

0.35

1=

0.95 =

wk = 0.6

1=

1=

0.43

0.4

wk = 0.35

0.65 =

1=

0.05

k J > (a3, a 4 )

0.4

0.4

wk = 0.57

k J > (a2, a 4 )

k J < (a3, a 4 )

wk =

wk = 0.55

wk = 0.6

wk = 0.6

0.35 0.95 =

0.6 = 0

k J > (a2, a3)

wk

0.95 =

wk = 0.4

k J > (a1, a4 )

k J (a2, a 4 )

k J < (a2, a 4 )

w2

wk

k J (a2, a3)

k J < (a2, a3)

w1 + w 2 w2

wk = 0.6

k J (a1, a 4 )

k J < (a1, a 4 )

wk = 0.6

k J > (a2, a1)

k J < (a3, a1)

w1 + w2 w2

wk

wk =

0.65

0.65

Computers & Industrial Engineering 137 (2019) 106067

X. Liu and S.-p. Wan

Combining all the inequalities above, Inequalities (3-7) can be obtained as follows:

w1 0.43 w2 0 w1 + w 2 w2 w1

0.35 0.17

Table 6 Concordance index.

a1 a2 a3 a4

(3-7)

Moreover, w1 and w2 must satisfy

0 < 0.35 + w1 < 1 0 < 0.37 + w2 < 1 w1 + w 2 = 0

(3-8)

a1

a2

a3

a4

× 0.95 0.61 0.54

0.39 × 0.58 0.34

0.42 1 × 0.34

0.95 1 0.66 ×

a1

a2

a3

a4

× 0.95 0.59 0.56

0.41 × 0.56 0.36

0.44 1 × 0.36

0.95 1 0.64 ×

Table 7 Concordance index.

According to Inequalities (3-7) and (3-8), the following inequalities are established.

w1 0.43 w2 0 w2 w1 0.17 0.35 < w1 < 0.65 0.37 < w2 < 0.63 w1 + w 2 = 0

a1 a2 a3 a4

It follows that

0

0.35 < w1 0.65 w2 < 0.63 2 w1 0.17 w2 = w1

Table 8 Decision matrix.

Thus, we have

a1 a2 a3 a4 a5 a6

0.35 < w1 0.085 0 w2 < 0.63 w2 = w1 Therefore, the ranges of w1 and w2 are obtained as

0

0.35 < w1 0 w2 < 0.35 w2 = w1

(3-9)

c1

c2

c3

c4

c5

c6

c7

c8

c9

c10

c11

80 55 83 40 52 94

500 580 600 450 880 960

1000 250 450 1000 900 950

5.2 3 3.8 7.5 3 3.6

8 1 4 7 3 5

0.5 4 3.5 0 4.5 3.5

9 3 7 10 2 4

0 5 65 0 10 10

2 8 6 10 5 3

300 175 125 450 150 250

4200 900 850 900 750 2000

Therefore, all outranking relations for each pair of alternatives are unchanged in this case. Case 2: Let w1 = 0.01 and w2 = 0.01, i.e., w1 and w2 dissatisfy Inequalities (3-9), implying that w1 and w2 are not within the ranges derived from Inequalities (3-9). The weights of criteria c4 and c5 are changed as 0.36 and 0.36, respectively. Moreover, the new concordance indices are obtained in Table 7. The outranking relations are constructed as follows:

In the following, the outranking relations for the example above are verified under the conditions that Inequalities (3-9) are either satisfied or not satisfied. Case 1: Let w1 = 0.01 and w2 = 0.01, i.e., w1 and w2 satisfy Inequalities (3-9), implying that w1 and w2 are within the ranges derived from Inequalities (3-9). The weights of criteria c4 and c5 are changed as 0.34 and 0.38, respectively. Moreover, the new concordance indices of this example are derived in Table 6. As the discordance indices D (ai , aj ) does not depend on the criteria weights, D (ai , aj ) remains unchanged. Therefore, the outranking relations are built as follows:

6

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It is easy to discover that the outranking relation of alternatives a1 and a3 is changed. It is reliable and effective to analyze the sensitivity of criteria weights in ELECTRE I.

Table 11 Discordance index. a1

4. Example 2 for ELECTRE I To facilitate the understanding of the process of sensitivity analysis for ELECTRE I, another example is presented in this section. This example was discussed in Leyva-López and Fernández-González (2003). An electricity power plant has to be built in the European Union. The problem is to identify the best possible location. The country members of the Union introduced six possible locations:

a1

×

a2 a3

1 1

a4

1

a5

1

a6

1

a2 80 3300

× 50 200 130 750 150 650 5 1100

a3

a4

a5

a6

380 3450

460 2200

1

100 3350

150 3300

150 550 100 450 55 1150

×

1

300 430 200 1100

×

1

2 1250

×

1 ×

1 1

1 1

1 1

a1:Italy a2 :Belgium a3 :Germany a4 :Sweden a5:Austria a6:France For this application, the following 11 criteria are formulated:

c1:Manpower c3:Construction cost c5:Villages to evaluate c7:Security level c9:Social impact c11:Financial return

c2: c4: c6: c8: c10:

Power in MW Annual maintenance cost Danger for environment CO emission Transport facilities to the plant

= 0.52 [0.5, 1) and the discordance threshold value = 0.125 (0, 1) . In fact, one can randomly choose a value from [0.5, 1) as the concordance threshold value, and randomly choose a value from (0, 1) as the discordance threshold value. Without loss of generality, we assumed that the weights of criteria c1 and c2 both change from 0.1, 0.2 to 0.1 + w1, 0.2 + w2 , where w1 and w2 satisfy 0 < 0.1 + w1 < 1, 0 < 0.2 + w2 < 1 and w1 + w2 = 0 , respectively. Next, to obtain the ranges of w1, w2 while keeping all the outranking relations unchanged, the following steps must be performed. Step 1: Calculate the concordance indices C (ai , aj ) and discordance j . The concordance indices D (ai , aj ) for i , j = 1, 2, 3, 4, 5, 6 and i and discordance indices of this example are calculated in Tables 10 and 11, respectively. Step 2: Build the outranking relations for each pair of alternatives by comparing these indices with the concordance threshold value = 0.52 and discordance threshold value = 0.125. It is easy to find from Table 11 that the outranking relations of pairs (a1, a6), (a2 , a1), (a2 , a3), (a2 , a4 ), (a2 , a5), (a2 , a6), (a3, a1), (a3 , a2 ), (a3, a4 ), (a3, a5), (a3, a6), (a4 , a1), (a4 , a2), (a4 , a3), (a4 , a5), (a4 , a6), (a5, a1), (a5, a2), (a5, a3), (a5 , a4 ), (a5 , a6), (a6 , a1), (a6 , a4 ) are not considered in this example, because their discordance indices are strictly greater . Hence, only the outranking relations of pairs than (a1, a2 ), (a1, a3), (a1, a4 ), (a1, a5), (a6, a2), (a6 , a3), (a6, a5) must be considered. According to the two rules of ELECTRE I in Section 2, the outranking relations can be built as follows:

The performance values of the alternatives on the criteria are presented in Table 8 and the criteria weights are shown in Table 9. In this example, we set the concordance threshold value Table 9 Weights of criteria. Criteria

c1

c2

c3

c4

c5

c6

c7

c8

c9

c10

c11

Weight

0.1

0.2

0.04

0.06

0.05

0.03

0.02

0.1

0.1

0.2

0.1

Table 10 Concordance index.

a1 a2 a3 a4 a5 a6

a1

a2

a3

a4

a5

a6

× 0.43 0.53 0.52 0.43 0.53

0.57 × 0.57 0.57 0.48 0.87

0.47 0.43 × 0.57 0.47 0.72

0.62 0.53 0.43 × 0.43 0.53

0.57 0.58 0.53 0.57 × 0.87

0.47 0.13 0.31 0.47 0.23 ×

7

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Step 3: Obtain the ranges of w1 and w2 while keeping the seven outranking relations above unchanged. According to Inequalities (2-1) and (2-4), the seven outranking relations above are not changed if and only if the following inequalities hold.

a6 a5 w1 + w 2

k J (a6, a5)

w1 + w 2

k J < (a6, a5)

=

a1 a2 w1 w1

k J (a1, a2)

w2

k J

=

<

wk = 0.52

(a1, a2)

wk

0.57 = k J

>

0.05

(a1, a2)

wk = 0.43

k J (a1, a4) k J < (a1, a4)

=

wk = 0.52

wk

0.62 =

k J > (a1, a4)

0.1

wk = 0.52

0.62

0.1

a1 a5 w1 w1

k J (a1, a5)

w2

wk = 0.52

0.57 =

wk

k J > (a1, a5)

k J < (a1, a5)

=

wk = 0.43

0.57

w1 + w 2

k J < (a6, a2)

=

wk

wk = 0.52

0.87 =

k J > (a6, a2)

0.87

a6 a3 k J (a6, a3)

w1 + w 2

k J < (a6, a3)

=

wk

wk = 0.52

0.72 =

k J > (a6, a3)

( w1 + w2)

0.2

wk = 0.31

(4-10)

w1, w2 can be ob-

In this section, ELECTRE II is reviewed and its sensitivity analysis is presented. In particular, the symbols used in this section are exactly the same as those in Section 2. In general, ELECTRE II comprises two stages. The first stage is the construction of the outranking relations for alternatives and the second stage is to obtain the final ranking. It is noteworthy that the construction of the outranking relations in ELECTRE II is similar to that in ELECTRE I. For each pair of alternatives (ai , aj ) , the concordance index C (ai , aj ) and the discordance index D (ai , aj ) are still calculated as follows:

0.74

w1 + w 2

0.87

5. Sensitivity analysis of ELECTRE II

0.35

wk = 0.13

wk = 0.23

0.07 w1 < 0.2 0.2 < w2 0.07 w1 = w2.

a6 a2 k J (a6, a2)

0.11

By solving Inequalities (4-10), the ranges of tained as follows:

0.05

0.14

w1 + w 2

k J > (a6, a5)

0.35

0.64

w1 + w 2 0.1 w1 0.05 w1 w2 0.14 |( w1 + w2) + 0.01| > 0 < 0.1 + w1 < 1 0 < 0.2 + w2 < 1 w1 + w 2 = 0

a1 a4 w1 + w 2

wk

0.87 =

0 < 0.1 + Combing above inequalities and w1 < 1, 0 < 0.2 + w2 < 1, w1 + w2 = 0 , the following inequalities are established.

0.57

0.14

w1 + w 2

wk = 0.52

0.72

0.41

C (ai , aj ) =

wk k J (ai, aj )

8

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and

D (ai , aj ) =

max{max{gk (aj )

gk (ai )}, 0}

k J

max{|gk (ai ) k J

gk (aj )|}

D (ai , aj ) . This implies that D (ai , aj ) < 1. According to Theorem > . Hence, both aj F ai and aj f ai cannot be 2.1, D (aj , ai ) = 1 > established. □

.

Therefore, if D (ai , aj ) > , then . That is, the pair (ai , aj ) does not need to be considered in this case. If D (ai , aj ) , according to Theorem 5.2, the pair (aj , ai ) does not need to be considered, because holds regardless of how the criteria weights change. If holds when D (ai , aj ) , then the following theorem is satisfied.

However, the definition of the outranking relations for ELECTRE II is different from that of ELECTRE I. ELECTRE II comprises two types of outranking relations (the strong and weak outranking relations). These relations are established by comparing the concordance indices and discordance indices with two pairs of threshold values: ( , ) and ( , ) , where 0.5 < < 1. The pair ( , ) < < 1 and 0 < is defined as the concordance and discordance thresholds for the strong outranking relations. Another pair ( , ) is defined as the thresholds for the weak outranking relations. Specifically, the outranking relations for ELECTRE II are built by the following rules:

, D (a i , a j ) (1) If C (ai , aj ) and C (ai , aj ) C (aj , ai ) , then alternative ai is regarded as strong outranking aj , denoted by ai , D (ai , aj ) (2) If C (ai , aj ) and C (ai , aj ) C (aj , ai ) , then alternative ai is regarded as weak outranking aj , denoted by ai

Theorem 5.3. For each pair of alternatives (ai , aj ) , if D (ai , aj ) , then 1 C (aj , ai )|) . holds if and only if C (ai , aj ) < 2 ( + C (aj , ai ) + | Proof. According to the rules of strong and weak outranking relations in Section 5, the outranking relation ai F aj does not hold if and only if one of the following conditions holds: (i) C (ai , aj ) < ; (ii) C (ai , aj ) < C (aj , ai ) ; (iii) D (ai , aj ) > .

the Fa ; j the fa . j

Similarly, the outranking relation ai one of the following conditions holds:

In the following, the strong and weak outranking relations are analyzed. Subsequently, the ranges of criteria weights are researched when the strong and weak outranking relations of two arbitrary alternatives remain unchanged. Each pair of alternatives (ai , aj ) satisfies at most one of the strong and weak outranking relations. Particularly, if the pair (ai , aj ) neither satisfies the strong outranking relation nor the weak outranking relation, then such a relation can be denoted by . Therefore, the following theorem is established. Theorem 5.1. For each pair of alternatives (ai , aj ), ai only if one of the following conclusions holds:

< D (a i , a j ) (i) if , then C (ai , aj ) C (a i , a j ) < (ii) if D (ai , aj ) , then

fa

j

and C (ai , aj ) and C (ai , aj )

Owing to D (ai , aj )

holds if and

C (aj , ai ) ; C (aj , ai ) .

1

C (ai , aj ) < 2 ( proof. □

j

holds, we have D (ai , aj )

C (aj , ai )|) . This completes the

+ C (aj , ai ) + |

By Theorem 5.3, holds if and only if D (ai , aj ) > or 1 C (ai , aj ) < 2 ( + C (aj , ai ) + | C (aj , ai )|) holds. If D (ai , aj ) > , then holds regardless of how the criteria weights change. This implies that such situation need not be considered. If D (ai , aj ) , then to maintain holding is equivalent to that the inequality 1 C (ai , aj ) < 2 ( + C (aj , ai ) + | C (aj , ai )|) holds. Based on the analysis of strong and weak outranking relations, keeping nine situations of the outranking relations for two arbitrary alternatives ai and aj unchanged is transformed into keeping the following three situations unchanged: (1) ai (2) ai (3)

Fa ; j fa ; j

holds when D (ai , aj )

.

Without loss of generality, we assume that the weights w1, w2, , wr (2 r n ) are changed as w1 + w1, w2 + w2, , wr + wr , respectively. Obviously, must satisfy w1, w2, , wr w1 + w2 + + wr = 0 and wk ( wk , 1 wk ) for k = 1, 2, , r , respectively. In the following, sensitivity analysis of criteria weights for

Theorem 5.2. For two arbitrary alternatives ai and aj , if one of the outranking relations ai F aj or ai f aj holds, then both aj F ai and aj f ai cannot be established. fa

, the following cases must be discussed:

In summary, when holds if and only if C (ai , aj ) < C (aj , ai ) C (ai , aj ) < or holds, i.e., C (ai , aj ) < max{ , C (aj , ai )} . It is noteworthy that 1 max{ , C (aj , ai )} = 2 ( + C (aj , ai ) + | C (aj , ai )|). Therefore,

The possible outranking relations for two arbitrary alternatives ai and aj are highly complicated. In fact, based on Theorem 2.1, some situations can be simplified by the following theorems.

or ai

does not hold if and only if

< D (a i , a j ) (1) when , the outranking relation ai F aj does not hold. Therefore, holds if and only if ai f aj does not hold. In this case, holds if and only if C (ai , aj ) < or C (ai , aj ) < C (aj , ai ) holds; (2) when D (ai , aj ) , the outranking relation ai F aj does not hold if and only if C (ai , aj ) < or C (ai , aj ) < C (aj , ai ) . However, ai f aj C (a i , a j ) < still establishes under the conditions that and C (ai , aj ) > C (aj , ai ) . Hence, ai F aj and ai f aj do not hold simultaneously if and only if at least one of the inequalities C (ai , aj ) < and C (ai , aj ) < C (aj , ai ) is established.

To keep ai f aj unchanged, according to Theorem 5.1, we have < D (a i , a j ) C (ai , aj ) and C (ai , aj ) C (aj , ai ) when , or C (a i , a j ) < and C (ai , aj ) C (aj , ai ) when D (ai , aj ) . Considering the outranking relation of the pair (ai , aj ) , according to Theorem 5.1 and the rules of strong and weak outranking relations, three situations may occur: ai F aj , ai f aj , and . Similarly, the pair (aj , ai ) also comprises three situations (aj F ai , aj f ai , and ). Hence, nine situations of the relations between ai and aj may occur:

Fa j

j

(a) C (ai , aj ) < ; (b) C (ai , aj ) < C (aj , ai ) ; (c) D (ai , aj ) > .

Proof. According to the rules of the outranking relations for ELECTRE II in Section 5, if ai f aj , then D (ai , aj ) . Therefore, there are following two cases need to be considered. < D (a i , a j ) Case 1: If , then ai f aj holds if and only if C (ai , aj ) and C (ai , aj ) C (aj , ai ) . Case 2: If D (ai , aj ) , then ai f aj holds if and only if C (a i , a j ) < and C (ai , aj ) C (aj , ai ) . Therefore, the conclusions of this theorem are true. □

Proof. Because ai

fa

or 9

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X. Liu and S.-p. Wan

ELECTRE II is discussed. The aim is to obtain the ranges of w1, w2, , wr while keeping the three outranking relations (ai F aj , ai f aj , and when D (ai , aj ) ) unchanged for each pair of alternatives (ai , aj ) . (a i , a j ) , For each pair of alternatives let Jl Jg {1, 2, , r } J > (ai , aj ), Je {1, 2, , r } J = (ai , aj ) , and {1, 2, , r } J < (ai , aj ) . If the pair (ai , aj ) satisfies ai F aj , then the following theorem is established. Theorem 5.4. The strong outranking relation ai and only if w1, w2, , wr satisfy

wk

k Jg Je k Jg

k J (ai, aj )

wk

wk

k Jl

Fa j

Case 2: When D (ai , aj ) C (aj , ai ) satisfy

C (a i , a j ) C (a i , a j ) < C (a i , a j )

wk

remains unchanged if

k J > (ai, aj )

wk

Theorem 5.6. If D (ai , aj ) if w1, w2, , wr satisfy

(5-11)

|

(5-12)

C (aj , ai ) =

k J (ai, aj )

wk +

k Jg Je

wk

k J (ai, aj )

wk +

k Je Jl

wk

(5-13)

k Jg Je

wk wk

k Jg

k J (ai, aj ) k Jl

(2) when D (ai , aj )

wk

k J

(ai, aj )

, then w1, w2,

k Jg Je

wk

k J (ai, aj )

wk

k Jg Je

wk <

k J (ai, aj )

wk

k Jg

wk

k Jl

wk

k J < (ai, aj )

wk

, wr satisfy

k J > (ai, aj )

wk

k J > (ai, aj )

wk

k J (ai, aj )

wk +

k Jg Je

wk

C (aj , ai ) =

k J (ai, aj )

wk +

k Je Jl

wk

(5-15)

(5-16)

Although the discordance index D (ai , aj ) does not depend on the criteria weights, it may affect the range of C (ai , aj ) . Hence, by Theorem 5.1, two cases must be analyzed: < D (a i , a j ) , ai f aj holds if and only if Case 1: When C (ai , aj ) and C (aj , ai ) satisfy

C (aj , ai )

+ C (aj , ai ) + |

C (aj , ai )|)

(5-20)

C (a i , a j ) =

k J (ai, aj )

wk +

k Jg Je

wk

C (aj , ai ) =

k J (ai, aj )

wk +

k Je Jl

wk

(5-21)

Table 12 Decision matrix.

C (a i , a j ) C (a i , a j )

1 ( 2

In this section, an example of sensitivity analysis of ELECTRE II is presented; Subsequently, the ranges are calculated for some criteria weights that are changed. Next, the strong and weak outranking relations are tested and verified for each pair of alternatives when the criteria weights are within the ranges. Finally, the change in the outranking relations is exhibited when the criteria weights are not in those ranges. This example adapted from Wang and Triantaphyllou (2008) is to choose the best position from four alternatives ai (i = 1, 2, 3, 4) for a waste incineration strategy in the eastern Switzerland region. Seven criteria ck (k = 1, 2, , 7) are considered in this decision. Without loss of generality, it is assumed that all the criteria are benefit criteria, i.e., the higher the score, the better the performance. In this example, the decision matrix is constructed in Table 12. The criteria weights are assumed in Table 13. Without loss of generality, we assumed that the weights of criteria c1

Proof. Obviously, the concordance indices C (ai , aj ) and C (aj , ai ) can be computed as follows:

C (a i , a j ) =

(5-19)

6. Example 1 of sensitivity analysis of ELECTRE II

(5-14)

, wr satisfy

wk

wk.

k Je Jl

By solving Inequalities (5-11), (5-14), (5-15), and (5-19) for all pairs of alternatives (ai , aj ) that satisfy ai F aj , ai f aj or (D (ai , aj ) ), the ranges of w1, w2, , wr can be obtained. Typically, it is easy to solve Inequalities (5-11), (5-14), (5-15) and (5-19) while the number of alternatives is not large. If these inequalities are highly complex or the number of alternatives is large, the Khachian algorithm (Khachian, 1979) also can be used to solve them and can only be obtained the values of w1, w2, , wr , but not the ranges.

wk <

wk

Combining Inequalities (5-20) and Eqs. (5-21), Inequalities (5-19) can be obtained. □

Theorem 5.5. The weak outranking relation ai f aj remains unchanged if and only if one of the following conditions holds: , then w1, w2,

k J (ai, aj )

k J (ai, aj )

wk

Obviously, C (ai , aj ) and C (aj , ai ) are computed as follows:

Without loss of generality, suppose that the outranking relation of the pair (ai , aj ) is ai f aj . Subsequently, the following theorem is established.

< D (a i , a j )

remains unchanged if and only

wk | > 2

k Je Jl

wk

k Jg Je

C (ai , aj ) <

Combining Inequalities (5-12) and Eqs. (5-13), Inequalities (5-11) can be obtained. □

(1) when

wk

, then

and

Proof. Because D (ai , aj ) holds, by Theorem 5.3, to keep unchanged, C (ai , aj ) and C (aj , ai ) need to satisfy

The concordance indices C (ai , aj ) and C (aj , ai ) are calculated as follows:

C (a i , a j ) =

k J (ai, aj )

+2

C (a i , a j ) C (aj , ai )

(5-18)

For each pair of alternatives (ai , aj ) , suppose that D (ai , aj ) hold. Therefore, the following theorem is proven.

Proof. As D (ai , aj ) does not depend on the criteria weights, the inequality D (ai , aj ) always holds. To keep ai F aj unchanged, C (ai , aj ) and C (aj , ai ) need to satisfy

C (a i , a j )

C (aj , ai )

Combing Eqs. (5-16) and Inequalities (5-18), Inequalities (5-15) can be obtained. □

wk

k J < (ai, aj )

holds if and only if C (ai , aj ) and

fa j

, ai

a1 a2 a3 a4

(5-17)

Combing Eqs. (5-16) and Inequalities (5-17), Inequalities (5-14) can be obtained. 10

c1

c2

c3

c4

c5

c6

c7

125 11980 31054 28219

866 900 883 840

9.81 11.45 9.86 10.38

218 189 172 171

1.41 1.45 1.82 1.95

542 452 341 339

483 303 311 318

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X. Liu and S.-p. Wan

outranking relations above unchanged. Based on Inequalities (5-11), (514), (5-15) and (5-19), the three relations above remain unchanged if and only if w1 and w2 simultaneously satisfy the following three groups inequalities.

Table 13 Weights of criteria. Criteria

c1

c2

c3

c4

c5

c6

c7

Weight

0.38

0.42

0.04

0.06

0.05

0.03

0.02

(1) a3

and c2 are changed as 0.38 + w1, 0.42 + w2 , respectively, where w1 w2 satisfy 0 < 0.38 + w1 < 1, 0 < 0.42 + w2 < 1, and and w1 + w2 = 0 . To obtain the ranges of w1 and w2 while keeping all the outranking relations unchanged, the following steps must be completed. Step 1: Calculate the concordance indices and discordance indices for each pair of alternatives (ai , aj ) by formula

C (ai , aj ) =

fa 1

w1 + w 2

< D (a3, a1) <

k J (a3, a1)

w1 + w 2

k J < (a3, a1)

= 0.11 (2) a3

Fa

w1 + w 2 4

w1 + w 2

wk

0.89 =

k J (a3, a 4) k J

<

(a3, a4)

wk =

k J > (a3, a1)

wk

0.78

wk = 0.6

wk

0.39

k J

>

0.89 =

0.29

w = (a3, a4 ) k

0.78

wk k J (ai, aj )

Combing all the inequalities above, the following Inequalities (6-22) can be obtained.

and

D (ai , aj ) =

max{ max {gk (aj ) 1 k 7

max {|gk (ai )

1 k 7

gk (ai )}, 0} gk (aj )|}

w1 + w 2 0.29 | w1 0.05| > 3( w1

.

(6-22)

0.05)

It is noteworthy that w1 and w2 must satisfy

In this example, the concordance indices are computed in Table 14 and the discordance indices are calculated in Table 15. Step 2: Establish the strong and weak outranking relations for each pair of alternatives by comparing these indices with two types of threshold values ( , ) and ( , ). In this example, set = 0.0056, = 0.0069 . Subsequently, the out= 0.5, = 0.6 and ranking relations for (a3 , a1), (a3, a2 ) and (a3 , a4 ) are established as follows based on the two rules of strong and weak outranking relations in Section 5. It is noteworthy that the outranking relations of pairs (a1, a2), (a2 , a1), (a1, a3), (a1, a4 ), (a4 , a1), (a2 , a3), (a2 , a4 ), (a4 , a2), (a4 , a3) are not considered, because their discordance indices are strictly greater than .

0 < 0.38 + w1 < 1 0 < 0.42 + w2 < 1 w1 + w 2 = 0

(6-23)

Obviously, the inequality | w1 0.05| > 3( w1 0.05) is not held when w1 0.05 0 . Hence, by Inequalities (6-22) and (6-23), the following inequalities are established.

0.38 < w1 < 0.62 0.42 < w2 < 0.58 w1 0.05 < 0 w2 = w1 It follows that

0.38 < w1 < 0.05 058 < w1 < 0.42 w2 = w1 Therefore, the ranges of w1 and w2 can be obtained as follows:

0.38 < w1 < 0.05 0.05 < w2 < 0.38 w2 = w1

In the following, the strong and weak outranking relations for the example above are tested and verified under the conditions that Inequalities (6-24) are satisfied and not satisfied. Case 1: Let w1 = 0.04 and w2 = 0.04 , i.e., w1 and w2 satisfy Inequalities (6-24), implying that w1 and w2 are within the ranges derived from Inequalities (6-24). The weights of criteria c1, c2 are changed as 0.38 + w1 = 0.42, 0.42 + w2 = 0.38 , respectively. Moreover, the concordance indices of this example can be calculated in Table 16. Because the discordance indices D (ai , aj ) does not depend on the criteria weights, D (ai , aj ) remains unchanged. Thus, according to the two rules of strong and weak outranking relations for ELECTRE II, the

Step 3: Obtain the ranges of w1 and w2 while keeping the three Table 14 Concordance index.

a1 a2 a3 a4

a1

a2

a3

a4

× 0.89 0.89 0.47

0.11 × 0.45 0.45

0.11 0.55 × 0.11

0.53 0.55 0.89 ×

Table 15 Discordance index.

a1 a2 a3 a4

(6-24)

Table 16 Concordance index. a1

a2

a3

a4

× 0.1518 0.0065 0.0072

1 × 0.0058 0.0070

1 1 × 1

1 1 0.0025 ×

a1 a2 a3 a4

11

a1

a2

a3

a4

× 0.89 0.89 0.51

0.11 × 0.49 0.49

0.11 0.51 × 0.11

0.49 0.51 0.89 ×

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X. Liu and S.-p. Wan

Table 17 Concordance index.

a1 a2 a3 a4

Table 18 Decision matrix.

a1

a2

a3

a4

× 0.89 0.89 0.53

0.11 × 0.51 0.51

0.11 0.49 × 0.11

0.47 0.49 0.89 ×

a1 a2 a3 a4 a5

strong and weak outranking relations are established as follows:

c1

c2

c3

c4

c5

−14 129 −10 44 −14

90 100 50 90 100

0 0 0 0 0

40 0 10 5 20

100 0 100 20 40

c1:financial c2: solution delivery c3:strategic contribution c4: risk management c5:environmental The performance values of the alternatives on the criteria are presented in Table 18, and the criteria weights are shown in Table 19. = 0.55, = 0.65 and In this example, we set the threshold values = 0.3, = 0.65. In fact, one can randomly choose and from [0.5, 1) , and randomly choose and from (0, 1) . Without loss of generality, suppose that the weights of criteria c1, c2 , and c3 change from 0.2, 0.2, 0.2 to 0.2 + w1, 0.2 + w2 , and 0.2 + w3 , respectively, where w1, w2, w3 satisfy 0 < 0.2 + w1 < 1, 0 < 0.2 + w2 < 1, 0 < 0.2 + w3 <1 and w1 + w2 + w3 = 0 . Next, to obtain the ranges of w1, w2, w3 while keeping all the outranking relations unchanged, the following steps must be performed. Step 1: Calculate the concordance indices C (ai , aj ) and discordance j . The concordance and indices D (ai , aj ) for i , j = 1, 2, 3, 4, 5 and i discordance indices of this example are calculated in Tables 20 and 21, respectively. Step 2: Build the outranking relations for each pair of alternatives by comparing these indices with the threshold values = 0.55, = 0.65 and = 0.3, = 0.65. According to Table 21, it is obvious that the pairs (a1, a2), (a1, a4 ), (a2 , a1), (a2 , a3), (a3, a1),

Obviously, all outranking relations of each pair of alternatives are not changed in this case. Case 2: Let w1 = 0.06 and w2 = 0.06 , i.e., w1 and w2 dissatisfy Inequalities (6-24), implying that w1 and w2 are not within the ranges derived from Inequalities (6-24). The weights of criteria c1, c2 are changed as 0.38 + w1 = 0.44, 0.42 + w2 = 0.36 . Moreover, the concordance indices of this example can be calculated in Table 17. According to the two rules of the strong and weak outranking relations for ELECTRE II, the strong and weak outranking relations of this example are obtained as follows:

(a3, a2), (a3, a4 ), (a3, a5), (a4 , a1), (a4 , a2), (a4 , a3), (a5, a1), (a5, a2), (a5, a3), (a5, a4 )

are not considered in this example, because their discordance indices are strictly greater than . Hence, only the outranking relations of Table 19 Weights of criteria. c1

c2

c3

c4

c5

Weight

0.2

0.2

0.2

0.2

0.2

Table 20 Concordance index.

It is easy to discover that the outranking relation of alternatives a3 and a2 is changed. This demonstrates that the sensitivity analysis of criteria weights is reliable and effective for ELECTRE II.

a1 a2 a3 a4 a5

7. Example 2 for ELECTRE II To better understand the process of sensitivity analysis of ELECTRE II, another example is presented in this section. This example adapted from Buchanan, Sheppard, and Lamsade (1999) is to rank the Northern Generation projects, in which alternatives (i.e., projects) are clearly defined as follows:

a1:Penstock

Criteria

a1

a2

a3

a4

a5

× 0.6 0.6 0.6 0.6

0.6 × 0.2 0.6 0.8

0.8 0.6 × 0.6 0.6

0.8 0.6 0.6 × 0.8

0.8 0.6 0.4 0.4 ×

a2

a3

a4

a5

4 40 100 139

58 80 20 85 54 80

10 60 40 143 50 60 20 58

Table 21 Discordance index. a1

a2 : Power Station Area Rock

Stabilisation a3:Automatic Generator Control a4 : Lower Station Electrical Upgrade a5: Station Forced Ventilation. After discussing with the management team, the following five criteria were used to evaluate the alternatives. 12

a1

×

1

a2

100 143

×

a3

1

1

a4

1

1

1

a5

1

1

1

×

× 1

×

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X. Liu and S.-p. Wan

pairs (a1, a3), (a1, a5), (a2 , a4 ), (a2 , a5), (a4 , a5) must be considered. From the two rules of ELECTRE II in Section 5, it is easy to build the outranking relations of pairs (a1, a3), (a1, a5), (a2 , a4 ), (a2 , a5), (a4 , a5) as follows:

situations of the outranking relations unchanged when the sensitivity analysis of the criteria weights was performed. Subsequently, the ranges of criteria weights were presented by inequalities under the condition that the outranking relations for two arbitrary alternatives

Step 3: Obtain the ranges of w1, w2 and w3 keeping the five relations above unchanged. According to Inequalities (5-11), (5-15), and (5-19), the five relations above are not changed if and only if the following inequalities hold.

remain unchanged. For ELECTRE II, nine situations of the strong and weak outranking relations were simplified into three situations when the sensitivity analysis was performed. Subsequently, some inequalities were derived to calculate the ranges of criteria weights when the outranking relations between any two alternatives were unchanged. In addition, four examples were presented to verify the effectiveness of the sensitivity analyses of criteria weights in ELECTRE I and II, separately. However, the sensitivity of the weights and performance values in ELECTRE III were not discussed herein, but will be discussed in the future. In addition, it would be interesting to study how the rank reversals for ELECTRE methods can be fully resisted.

(1)a1

Fa 3

w 2 + w3 w2 w1

0.65 0.2

0.8 =

0.15

(2)a1

Fa 5

w1 + w3 0.65 w2 0.2

0.8 =

0.15

(3)a2

fa

w1 + w2 + w3 0.55 w1 + w2 + w3 < 0.65 w1 + w2 0

4

0.6 = 0.05 0.6 = 0.05

w1, w2, w3 It is noteworthy that satisfy 0 < 0.2 + w1 < 1, 0 < 0.2 + w2 < 1, 0 < 0.2 + w3 < 1, and w1 + w2 + w3 = 0 . Hence, after combining and solving all the inequalities above, the ranges of w1, w2, w3 can be obtained as follows:

Acknowledgments This research was supported by the National Natural Science Foundation of China (Nos. 11701236, 71740021, 11861034, 11461030 and 61263018), the Natural Science Foundation of Jiangxi Province of China (No. 20192BAB207012), “Thirteen five” Programming Project of Jiangxi Province Social Science (No. 18GL13).

0.2 < w1 0.15 0.15 < w2 0.15 0.2 < w3 0 w1 + w2 + w3 = 0.

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