2.1 Establishment of the Criteria System

The three pillars, i.e., economic prosperity, environmental protection, and social development, are a common ground of numerous sustainability standards; therefore, the criteria system for the sustainability assessment usually consists of the economic, environmental, and social factors. While the technological and political aspects can affect the economic, environmental, and social performances (Ren et al., 2013d), the technological and political issues are also usually incorporated in the criteria systems. In this study, a total of 14 criteria concerning the four aspects, i.e., economic, environmental, technological, and social–political issues (Acar and Dincer, 2014; Jayswal et al., 2011; Li et al., 2011; Liu and Huang, 2012; Lou et al., 2004; Martins et al., 2007; Othman et al., 2010; Pilavachi et al., 2009; Piluso et al., 2010; Ren et al., 2013a; Ruiz-Mercado et al., 2012; Sadhukhan and Ng, 2011; Schwarz et al., 2002; Sikdar, 2003, 2007), were used for the sustainability assessment of various industrial systems (Table 9.1).

2.2 Fuzzy AHP Method for Scoring the Soft Criteria

There are usually two ways for determining the relative priorities of the alternatives with respect to the soft criteria: the scaling system method (Manzardo et al., 2012; Othman et al., 2010) and the pairwise comparison method (Ren et al., 2014; Saaty, 1980). The scaling method scores the alternatives by using numbers (crisp numbers or gray numbers), while the pairwise comparison method determines the relative priorities of the alternatives via the pairwise comparison. The scaling method is simple and easy to be operated; however, it cannot assure the overall consistency among all the relative priorities of the alternatives with respect to each of the soft criteria. AHP is a widely used pairwise comparison method for determining the relative priorities of the alternatives with respect to the soft criteria (Ren et al., 2014; Saaty, 1980). However, the conventional AHP method uses nice scales (1, 2, …, 9) and their reciprocals to determine the comparison matrix and the relative priorities of the criteria, which requires the users to describe their opinions using crisp numbers. As human’s judgments are usually subjective, vague, and ambiguous, this limitation could result in the inconvenience of the users and inaccuracy of the results (Ren and Sovacool, 2014). Accordingly, this study adopted a fuzzy AHP method by combining the thoughts of the conventional AHP with the fuzzy set theory to quantify the relative priorities of the soft criteria. The fuzzy AHP method allows the stakeholders/decision-makers to describe their preferences by using linguistic variables, i.e., words or sentences in a natural or artificial languages, which is more suitable than the crisp numbers for depicting human’s judgments. The linguistic variables are in turn connected to fuzzy numbers through the membership functions (Afgan et al., 2008; Ren et al., 2015).

For a universe set, X, the fuzzy set A in X is characterized by a membership function ${\mu }_{\tilde{A}}(x)\to [0,1]$, which quantifies the grade of membership of the element X to A (Amindoust et al., 2012; Mazloumzadeh et al., 2008; Zadeh, 1965). The membership functions can be of different formulation, but in practice, triangular and trapezoidal membership functions are most frequently used in the fuzzy logic as showed in Eqs. (9.1) and (9.2) (Mazloumzadeh et al., 2008), in which a, b, c, and d are parameters. For those unfamiliar to the fuzzy set theory, it also has been extensive used in some other literatures (Amindoust et al., 2012; Mazloumzadeh et al., 2008; Zadeh, 1965).

${\mu }_{\tilde{A}}(x)=\{\begin{array}{ll}0\hfill & x\le a\hfill \\ \frac{x-a}{b-a}\hfill & a<x\le b\hfill \\ \frac{x-c}{b-c}\hfill & b<x\le c\hfill \\ 0\hfill & x>c\hfill \end{array}$ (9.1)

(9.1)

${\mu }_{\tilde{A}}(x)=\{\begin{array}{ll}0\hfill & x\le a\begin{array}{c}\mathrm{or}\begin{array}{c}x\ge d\end{array}\end{array}\hfill \\ \frac{x-a}{b-a}\hfill & a<x\le b\hfill \\ 1\hfill & b<x\le c\hfill \\ \frac{x-d}{c-d}\hfill & x>c\hfill \end{array}$ (9.2)

(9.2)

In the fuzzy AHP method, the comparison matrix is first determined by using the linguistic terms, which are then transformed into fuzzy numbers (Table 9.2) (Tseng et al., 2009). Assuming $X=\left\{x_1,x_2,\dots ,x_n\right\}$ is an object set, and $U=\left\{g_1,g_2,\dots ,g_m\right\}$ is a goal set, then, the performance of each object regarding each goal can be analyzed; the m-extent analysis values for each object can be obtained as the following equation:

$M_{gi}^1,M_{gi}^2,\dots ,M_{gi}^m,\begin{array}{c}i=1,2,\dots ,n\end{array}$ (9.3)

(9.3)

where $M_{gi}^j(j=1,2,\dots ,m)=(l_{gi}^j,m_{gi}^j,u_{gi}^j)$ are the triangular fuzzy numbers.

Subsequently, the fuzzy AHP method is conducted according to the followed four steps as developed by Chang (1996) (Choudhary and Shankar, 2012; Heo et al., 2010; Tseng et al., 2009).

Step 1: The value of the fuzzy synthetic extent with respect to the ith object is defined as the following equation:

$S_i=\sum _{j=1}^m{M}_{gi}^j\otimes {\left[\sum _{i=1}^n\sum _{j=1}^m{M}_{gi}^j\right]}^{-1}$ (9.4)

(9.4)

where ${\sum }_{j=1}^m{M}_{gi}^j$ and ${\left[{\sum }_{i=1}^n{\sum }_{j=1}^m{M}_{gi}^j\right]}^{-1}$ can be determined according to Eqs. (9.5) and (9.6), respectively.

$\sum _{j=1}^m{M}_{gi}^j=\left(\sum _{j=1}^m{l}_{gi}^j,\sum _{j=1}^m{m}_{gi}^j,\sum _{j=1}^m{u}_{gi}^j\right),i=1,2,\dots ,n$ (9.5)

(9.5)

${\left[{\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^m{M}_{gi}^j\right]}^{-1}=\left(\frac{1}{{\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^m{u}_{gi}^j},\frac{1}{{\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^m{m}_{gi}^j},\frac{1}{{\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^m{l}_{gi}^j}\right)$ (9.6)

(9.6)

Step 2: The degree of possibility of $S_2=\left(l_2,m_2,u_2\right)\ge S_1=\left(l_1,m_1,u_1\right)$ is defined as the following equation:

$\begin{array}{ll}\hfill V\left(S_2\ge S_1\right)& =\underset{y>x}{\sup }\left(\min \left\{{\mu }_{\tilde{A}}(x),{\mu }_{\tilde{B}}(y)\right\}\right)\\ & =\mathrm{height}\left(S_2\cap S_1\right)={\mu }_{S_2}(d)=\{\begin{array}{l}1\mathrm{if}\begin{array}{c}{m}_2\ge m_1\end{array}\hfill \\ 0\mathrm{if}\begin{array}{c}{l}_1\ge u_2\end{array}\hfill \\ \frac{l_1-u_2}{(m_2-u_2)-(m_1-l_1)}\hfill & \mathrm{otherwise}\hfill \end{array}\end{array}$ (9.7)

(9.7)

where d is the ordinate of the highest intersection point between ${\mu }_{M_1}$ and ${\mu }_{M_2}$ as illustrated in Fig. 9.2. Both $V\left(S_2\ge S_1\right)$ and $V\left(S_1\ge S_2\right)$ are necessary for comparing $S_1$ and $S_2$.

Step 3: The degree of possibility for a convex fuzzy number to be greater than the convex fuzzy number $S_i(i=1,2,\dots ,k)$ is defined by the following equation:

$\begin{array}{l}V\left(S\ge S_1,S_2,\dots ,S_k\right)\hfill \\ =V\left(S\ge S_1\right)\mathrm{and}\begin{array}{c}V\left(S\ge S_2\right)\end{array}\begin{array}{c}\mathrm{and}\dots \mathrm{and}\begin{array}{c}V\left(S\ge S_k\right)\end{array}\end{array}\hfill \\ =\min V\left(S\ge S_i\right)\begin{array}{c},i=1,2,\dots ,k\end{array}\hfill \end{array}$ (9.8)

(9.8)

Assuming that

$d\prime \left(A_i\right)=\min V\left(S_i\ge S_k\right),k=1,2,\dots ,n\begin{array}{c}\mathrm{and}\begin{array}{c}k\ne i\end{array}\end{array}$ (9.9)

(9.9)

Then, the weight vector of the n elements $A_i(i=1,2,\dots ,n)$ can be determined by the following equation:

$W\prime ={\left(d\prime \left(A_1\right),d\prime \left(A_2\right),\dots ,d\prime \left(A_n\right)\right)}^T$ (9.10)

(9.10)

where $d\prime \left(A_i\right)$ is the weight of the ith element $A_i$, and $W\prime$ is the weight vector.

Step 4: The weight vector in Eq. (9.10) is normalized to obtain the weight of each element according to Eqs. (9.11) and (9.12), respectively. In Eq. (9.11), W is a nonfuzzy number.

$W={\left(d\left(A_1\right),d\left(A_2\right),\dots ,d\left(A_n\right)\right)}^T$ (9.11)

(9.11)

$d(A_i)=\frac{d\prime \left(A_i\right)}{{\displaystyle \sum }_{i=1}^{n}d\prime \left(A_i\right)}$ (9.12)

(9.12)

After the normalization, the weights of the criteria satisfy the following equation:

${\displaystyle \sum }_{i=1}^{n}d\left(A_i\right)=1$ (9.13)

(9.13)

where $d\left(A_i\right)$ is the normalized weight of the ith element $A_i$, and $W$ is the normalized weight vector.

2.3 Fuzzy ANP Method

ANP is a widely used MCDM technique that was derived from the AHP method (Saaty, 1996). The original AHP method was developed by Saaty in the 1970s to rank the priority sequence associated with the alternatives of a specific problem in a ratio scale by combining the tangible and intangible aspects (Dagdeviren and Yuksel, 2010; Escobar and Moreno-Jimenez, 2002; Saaty, 1980). However, the method is unable to deal with any kinds of interdependence between the evaluation criteria; thus, it cannot be used to solve the decision-making problems that involve the interactions between the assessment factors (Xu and Chan, 2013). Accordingly, the ANP method was developed by extending the AHP technique to address the MCDM problems with interactions and interdependencies among the alternatives or criteria (Dagdeviren and Yuksel, 2010; Luo et al., 2010; Saaty, 1996; Xu and Chan, 2013). The structural difference between ANP and AHP (Fig. 9.3) is that AHP represents a framework with a unidirectional hierarchical relationship, while ANP uses a hierarchical networks with feedback approaches among decision levels and attributes (Atmaca and Basar, 2012; Dagdeviren and Yuksel, 2010). However, the conventional ANP method is based on the AHP method, it does not perform well when facing the subjectivity, vagueness, and ambiguity existed in human’s judgments as discussed above. Thus, the fuzzy ANP method was employed for determining the weights of the criteria in this study, and its procedure usually consists of four steps as showed in Fig. 9.4 (Atmaca and Basar, 2012; Lee et al., 2009a; Xu and Chan, 2013).

Step 1: Determining the evaluation network structure. According to the principle of ANP and the evaluation criteria, the evaluation network structure with the clarified relationships between the criteria or alternatives is to be established in this step.

Step 2: Establishing the fuzzy comparison matrices and calculating priority vectors. The fuzzy comparison matrices for pairwisely comparing the elements of each cluster are to be established in this step. After the matrices are established, the local priority vectors for each pairwise comparison matrix can be obtained by using the fuzzy AHP method (Chang, 1996). It is worth pointing out that the determination of the fuzzy comparison matrices should be based on a group decision-making, e.g., a focus group, a seminar, or a teleconference of the experts in the related areas including professors, senior engineers, government officials (investors and administrative executors), etc. During the group decision-making process, a consensus about the interdependencies and interactions among the criteria was achieved by exchanging opinions and discussion, which ensure the decision-making to incorporate the preferences/willingness of all the stakeholders and to avoid the inconsistencies existing in the opinions of the decision-makers.

Step 3: Calculating the supermatrix, weighted supermatrix formation, and limit supermatrix. In this step, the global priorities of a specific system with the interdependent factors are obtained by inputting the local priority vectors in the appropriate columns of a matrix. Assuming C_k to be the kth cluster (k=1, 2, …, N) that has nk elements (e_k1, e_k2, …, e_nk), the standard form of a supermatrix is as Eq. (9.14). In the supermatrix, the segment, W_ij, represents the relationship between the ith cluster and the jth cluster; each column of W_ij is the local priority vector obtained from the corresponding pairwise comparison matrix, indicating the relative importance of the elements in the ith cluster to an element in the jth cluster (Xu and Chan, 2013).

Subsequently, the pairwise comparisons are carried out between the clusters to determine the weight matrix, in which each column represents the relative effects between the other clusters on a specific cluster with respect to the corresponding criterion and the numerical sum of the elements in each column is 1. Then, according to the weight matrix, the supermatrix is normalized to obtain the weighted supermatrix, which is subsequently transformed into the limit supermatrix by raising it to powers for ensuring the convergence of the matrix (Xu and Chan, 2013).

$\begin{array}{ccccccc}& & C_1& \cdots & C_k& \cdots & C_N\\ & & e_{11}\cdots e_{1n_1}& \cdots & e_{k1}\cdots e_{k{n}_k}& \cdots & e_{n1}\cdots e_{n{n}_N}\\ C_1& \begin{array}{l}{e}_{11}\\ ⋮\\ e_{1n_1}\end{array}& W_{11}& \cdots & W_{1k}& \cdots & W_{1N}\\ ⋮& & ⋮& ⋮& ⋮& ⋮& ⋮\\ C_k& \begin{array}{l}{e}_{k1}\\ ⋮\\ e_{k{n}_1}\end{array}& W_{k1}& \cdots & W_{kk}& \cdots & W_{kN}\\ ⋮& & ⋮& ⋮& \ddots & ⋮& ⋮\\ C_N& \begin{array}{l}{e}_{n1}\\ ⋮\\ e_{n{n}_N}\end{array}& W_{N1}& \cdots & W_{Nk}& \cdots & W_{NN}\end{array}$ (9.14)

(9.14)

Step 4: Obtaining the final weights of each elements.

The elements in each row of the limit supermatrix are same, and the value indicates the relative importance of each element. Then, the relative weights of the elements can be obtained by normalizing the limit supermatrix.

2.4 PROMETHEE Method for the Sustainability Prioritization

The prioritization of alternative industrial systems is a typical MCDM problem, the stakeholders/decision-makers have to select the most sustainable scenario from a variety of pathways by considering multiple criteria. Among the various MCDM methods available in the literatures, e.g., technique for order preference by similarity to ideal solution (TOPSIS) (Gharakhlou et al., 2010), data envelopment analysis (Ren et al., 2013c), elimination Et Choix Traduisant la REalité (Cavallaro, 2010), gray relational analysis (GRA) (Manzardo et al., 2012), the method of PROMETHEE has the advantages of highly efficient and easy to be used and operated due to its lower level of complexity (Cavallaro, 2009; Parreiras and Vasconcelos, 2007).

PROMETHEE, developed by Brans and Vincke in 1980s to rank the alternatives using the outranking principle, is a collective name of the PROMETHEE family including PROMETHEE I (partial ranking), PROMETHEE II (complete ranking), PROMETHEE III (ranking based on intervals), PROMETHEE IV (continuous case), PROMETHEE V (PROMETHEE II and integer linear programming), PROMETHEE VI (weights of the criteria are intervals), and PROMETHEE GAIA (graphical representation of PROMETHEE). One of the main advantages of PROMETHEE over other outranking methods is that the decision-makers can easily understand the concepts and parameters inherent in the method, which makes it simpler to be operated and consequently increases the effectiveness (Silva et al., 2010). In this study, PROMETHEE II (complete ranking) was selected as the decision-making tool for prioritizing the sustainability sequence of alternative industrial systems.

In PROMETHEE II, a complete preorder (complete ranking) of the alternatives is obtained according to the calculated net flow from each alternative. The analysis procedures include three steps: (1) establishment of the decision-making matrix with the data of the alternatives with respect to each criterion and the weights of the criteria; (2) selection of a preference function, by which the difference between two alternatives regarding a criterion was transformed into a value between 0 and 1; (3) calculation of the net outranking flow of the alternatives with respect to each criterion as the preference index to prioritize the alternatives (Chou et al., 2007). Among the three steps, the selection of the preference function is very critical, it can significantly affect the accuracy of the sustainability assessment. There are usually six types of preference functions corresponding to the six types of generalized criteria (see Table 9.3) (Chou et al., 2007). Among them, the Gaussian type preference function was selected in this study due to its advantages of sensitive to small variations of the PROMETHEE II input parameters (Ren et al., 2014) and containing continuity (Albadvi et al., 2007). In the Gaussian criterion as showed in Table 9.3, δ_j is defined as the threshold value and could be obtained by calculating the average variance of the jth criterion, and d_j represents the numerical difference of two alternatives regarding to each criterion.

Table 9.3

Shape of the Six Types of Generalized Criteria (Albadvi et al., 2007)

Generalized Criterion Type	Preference Function P(d)
Type I: Usual criteria $P_j\left(d_j\right)=\{\begin{array}{l}\begin{array}{cc}0& d_j=0\end{array}\\ \begin{array}{cc}1& \left|d_j\right|>0\end{array}\end{array}$
Type II: U shape criterion $P_j\left(d_j\right)=\{\begin{array}{l}\begin{array}{cc}0& \left|d_j\right|\le q\end{array}\\ \begin{array}{cc}1& \left|d_j\right|>q\end{array}\end{array}$
Type III: V-shape criterion $P_j\left(d_j\right)=\{\begin{array}{ll}\frac{\left|d_j\right|}{t}\hfill & \left|d_j\right|\le t\hfill \\ 1\hfill & \left|d_j\right|>t\hfill \end{array}$
Type IV: Level criterion $P_j\left(d_j\right)=\{\begin{array}{ll}0\hfill & \left|d_j\right|\le q\hfill \\ \frac{1}{2}\hfill & q<\left|d_j\right|\le t\hfill \\ 1\hfill & \left|d_j\right|>t\hfill \end{array}$
Type V: V-shape criterion with indifference criteria $P_j\left(d_j\right)=\{\begin{array}{ll}0\hfill & \left|d_j\right|\le q\hfill \\ \frac{\left|d_j\right|-q}{t-q}\hfill & q<\left|d_j\right|\le t\hfill \\ 1\hfill & \left|d_j\right|>t\hfill \end{array}$
Type VI: Gaussian criterion $P_j\left(d_j\right)=\{\begin{array}{ll}0\hfill & d_j\le 0\hfill \\ 1-e^{-\left(d_j^2/2{\delta }_j^2\right)}\hfill & d_j>0\hfill \end{array}$

In the PROMETHEE II method, the multicriteria preference index for a pair of alternatives A_i and A_k is defined as the following equation:

$H\left(A_i,A_k\right)=\sum _{j=1}^n{w}_j{P}_j\left(A_i,A_k\right)$ (9.15)

(9.15)

where $P_j\left(A_i,A_k\right)$ is the preference function, $w_j$ is the weights of the jth criterion calculated by using the ANP method. The positive outranking flow and negative outranking flow of an alternative can be calculated by Eqs. (9.16) and (9.17), respectively.

${\varphi }^{+}\left(A_i\right)=\sum _{k=1}^{m}H\left(A_i,A_k\right)(i=1,2,\dots ,m)$ (9.16)

(9.16)

${\varphi }^{-}\left(A_i\right)=\sum _{k=1}^{m}H\left(A_k,A_i\right)(i=1,2,\dots ,m)$ (9.17)

(9.17)

Then, the net outranking flow of an alternative can be calculated by the following equation:

$\varphi \left(A_i\right)={\varphi }^{+}\left(A_i\right)-{\varphi }^{-}\left(A_i\right)(i=1,2,\dots ,m)$ (9.18)

(9.18)

As the net outranking flow indicates how much each alternative is preferred to the others, its value can be used to rank the alternatives, i.e., a higher value of the net outranking flow corresponds to a better alternative (Albadvi et al., 2007; Parreiras and Vasconcelos, 2007). The standards can be described as Eqs. (9.19)–(9.20).

$\mathrm{if}\begin{array}{cc}\varphi \left(A_i\right)\succ \varphi \left(A_k\right)& \mathrm{then}\end{array}{A}_i\succ A_K$ (9.19)

(9.19)

$\mathrm{if}\begin{array}{cc}\varphi \left(A_i\right)~\varphi \left(A_k\right)& \mathrm{then}\end{array}{A}_i~A_K$ (9.20)

(9.20)

where “$\succ$” represents “superior to,” ~ represents “no difference.”

It is noteworthy that the values of the net outranking flow for the alternatives with respect to each criterion should be first transformed into dimensionless values, and the linear transformation method proposed by Hajkowicz and Higgins was adopted in this study. Meanwhile, the 14 criteria can be divided into 2 categories according to their effects: benefit criteria and cost criteria. The greater the value of an industrial system with respect to a benefit criterion, the more superior the industrial system will be. On the contrary, the greater the value of an industrial system with respect to a cost criterion, the less superior the industrial system will be. Accordingly, resource availability, maturity, energy efficiency, exergy efficiency, technology innovation, social acceptability, effect on energy security, and policy applicability belong to the benefit criteria, and the other six criteria belong to the cost criteria. Both the benefit criteria and cost criteria can be normalized after the transformation. Moreover, the cost criteria can also be transformed into benefit criteria after this process.

2.5 Sensitivity Analysis Method

In order to identify the most critical and sensitive criteria that have a significant effect on the sustainability sequence and to investigate the influence of the interactions and interdependencies among the criteria on the results, a novel sensitivity analysis method was developed by extending the method proposed by Triantaphyllou and Sanchez (1997).

Assuming $A_j\succ A_i$ according to the PROMETHEE method, the decision-makers want to alter the current ranking by modifying the weight of one criterion. With the tth criterion as an example, Eqs. (9.21) and (9.22) can be obtained.

$\begin{array}{l}\varphi \left(A_i\right)-\varphi \left(A_j\right)=\left[{\varphi }^{+}\left(A_i\right)-{\varphi }^{-}\left(A_i\right)\right]-\left[{\varphi }^{+}\left(A_j\right)-{\varphi }^{-}\left(A_j\right)\right]\hfill \\ =\left[\sum _{k=1}^{m}H\left(A_i,A_k\right)-\sum _{k=1}^{m}H\left(A_k,A_i\right)\right]-\left[\sum _{k=1}^{m}H\left(A_j,A_k\right)-\sum _{k=1}^{m}H\left(A_k,A_j\right)\right]\hfill \\ =\left[\sum _{k=1}^m\sum _{d=1}^n{w}_d{P}_d\left(A_i,A_k\right)-\sum _{k=1}^m\sum _{d=1}^n{w}_d{P}_d\left(A_k,A_i\right)\right]-\left[\sum _{k=1}^m\sum _{d=1}^n{w}_d{P}_d\left(A_j,A_k\right)-\sum _{k=1}^m\sum _{d=1}^n{w}_d{P}_d\left(A_k,A_j\right)\right]\hfill \\ =\left[\sum _{k=1}^m\left(\sum _{d=1,d\ne t}^n{w}_d{P}_d\left(A_i,A_k\right)+w_t{P}_t\left(A_i,A_k\right)\right)-\sum _{k=1}^m\left(\sum _{d=1,d\ne t}^n{w}_d{P}_d\left(A_k,A_i\right)+w_t{P}_t\left(A_k,A_j\right)\right)\right]\hfill \\ -\left[\sum _{k=1}^m\left(\sum _{d=1,d\ne t}^n{w}_d{P}_d\left(A_j,A_k\right)+w_t{P}_t\left(A_j,A_k\right)\right)-\sum _{k=1}^m\left(\sum _{d=1,d\ne t}^n{w}_d{P}_d\left(A_k,A_j\right)+w_t{P}_t\left(A_k,A_j\right)\right)\right]\hfill \\ =\sum _{k=1}^m\left[(w_t{P}_t\left(A_i,A_k\right)+w_t{P}_t(A_k,A_j))-(w_t{P}_t\left(A_j,A_k\right)+w_t{P}_t(A_k,A_i))\right]\hfill \\ +\sum _{k=1}^m\left(\sum _{d=1,d\ne t}^n{w}_d{P}_d\left(A_i,A_k\right)+w_d{P}_d\left(A_k,A_j\right)-w_d{P}_d\left(A_k,A_i\right)-w_d{P}_d\left(A_j,A_k\right)\right)\hfill \\ =w_t\left[\sum _{k=1}^m\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]\hfill \\ +\sum _{k=1}^m\left[\sum _{d=1,d\ne t}^n{w}_d{P}_d\left(A_i,A_k\right)+w_d{P}_d\left(A_k,A_j\right)-w_d{P}_d\left(A_k,A_i\right)-w_d{P}_d\left(A_j,A_k\right)\right]\ge 0\hfill \end{array}$ (9.21)

(9.21)

$\begin{array}{l}\sum _{k=1}^m\left[\sum _{d=1,d\ne t}^n{w}_d{P}_d\left(A_i,A_k\right)+w_d{P}_d\left(A_k,A_j\right)-w_d{P}_d\left(A_k,A_i\right)-w_d{P}_d\left(A_j,A_k\right)\right]\hfill \\ =\varphi \left(A_i\right)-\varphi \left(A_j\right)-w_t\left[\sum _{k=1}^m\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]\hfill \end{array}$ (9.22)

(9.22)

Assuming that for reversing the current ranking of $A_i$ and $A_j$, the minimum quantity of the weight of the tth criterion that has to be changed is ${\delta }_{t,i,j}$; then, the modified weight of the tth criterion, $w_t^{*}$, is

$w_t^{*}=w_t-{\delta }_{t,i,j}\ge 0$ (9.23)

(9.23)

In order to satisfy Eq. (9.13), the weight should be renormalized according to Eqs. (9.24) and (9.25).

${w\prime }_t=\frac{w_t^{*}}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}$ (9.24)

(9.24)

${w\prime }_d=\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}$ (9.25)

(9.25)

where ${w\prime }_t$ represents the new normalized weight of the tth criterion, and ${w\prime }_d\left(d=1,2,\dots ,n,d\ne t\right)$ represents the new normalized weight of the dth criterion.

According to Eq. (9.21), the new relationship between the net outranking flow of the ith alternative and that of the jth alternative is as Eq. (9.26), in which ${\varphi }^{*}\left(A_i\right)$ and ${\varphi }^{*}\left(A_j\right)$ represent the net flow of the ith alternative and the jth alternative, respectively.

In order to reverse the ranking of $A_i$ and $A_j$, ${\varphi }^{*}\left(A_i\right)-{\varphi }^{*}\left(A_j\right)$ should be greater than 0 according to Eq. (9.19). Accordingly, Eq. (9.27) should be satisfied.

$\begin{array}{c}{\varphi }^{*}\left(A_i\right)-{\varphi }^{*}\left(A_j\right)=\frac{w_t^{*}}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}\left[{\displaystyle \sum }_{k=1}^m\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]\\ +{\displaystyle \sum }_{k=1}^m\left(\begin{array}{l}{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_i,A_k\right)+{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_k,A_j\right)\\ -{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_k,A_i\right)-{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_j,A_k\right)\end{array}\right)\\ =\frac{w_t^{*}}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}\left[{\displaystyle \sum }_{k=1}^m\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]\\ +{\displaystyle \sum }_{k=1}^m\left(\begin{array}{l}{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_i,A_k\right)+{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_k,A_j\right)\\ -{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_k,A_i\right)-{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_j,A_k\right)\end{array}\right)\end{array}$ (9.26)

(9.26)

$\begin{array}{l}{\varphi }^{*}\left(A_i\right)-{\varphi }^{*}\left(A_j\right)=\frac{w_t^{*}}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}\left[{\displaystyle \sum }_{k=1}^m\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]\\ +{\displaystyle \sum }_{k=1}^m\left(\begin{array}{l}{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_i,A_k\right)+{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_k,A_j\right)\\ -{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_k,A_i\right)-{\displaystyle \sum }_{d=1,d\ne t}^n\frac{w_d}{w_t^{*}+{\displaystyle \sum }_{d=1,d\ne t}^n{w}_d}{P}_d\left(A_j,A_k\right)\end{array}\right)\ge 0\end{array}$ (9.27)

(9.27)

Multiply Eq. (9.27) by $w_t^{*}+{{\displaystyle \sum }}_{d=1,d\ne t}^n{w}_d$, we could obtain Eq. (9.28).

$\begin{array}{c}{\varphi }^{*}\left(A_i\right)-{\varphi }^{*}\left(A_j\right)=w_t^{*}\left[{\displaystyle \sum }_{k=1}^m\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]\\ +{\displaystyle \sum }_{k=1}^m\left({\displaystyle \sum }_{d=1,d\ne t}^n{w}_d{P}_d\left(A_i,A_k\right)+w_d{P}_d\left(A_k,A_j\right)-w_d{P}_d\left(A_k,A_i\right)-w_d{P}_d\left(A_j,A_k\right)\right)\ge 0\end{array}$ (9.28)

(9.28)

By integrating Eq. (9.28) with Eq. (9.22), Eq. (9.29) can be obtained.

${\varphi }^{*}\left(A_i\right)-{\varphi }^{*}\left(A_j\right)=\varphi \left(A_i\right)-\varphi \left(A_j\right)-\left(w_t-w_t^{*}\right)\times \left[{\displaystyle \sum }_{k=1}^m\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]$ (9.29)

(9.29)

Similarly, Eq. (9.30) can be obtained by integrating Eqs. (9.29) and (9.23).

${\varphi }^{*}\left(A_i\right)-{\varphi }^{*}\left(A_j\right)=\varphi \left(A_i\right)-\varphi \left(A_j\right)-{\delta }_{t,i,j}\times \left[{\displaystyle \sum }_{k=1}^m\left(P_t(\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]$ (9.30)

(9.30)

Finally, Eq. (9.31) can be obtained.

${\delta }_{t,i,j}\left[{\displaystyle \sum }_{k=1}^m\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]\le \varphi (A_i)-\varphi \left(A_j\right)$ (9.31)

(9.31)

According to Eq. (9.31), if $\begin{array}{c}{{\displaystyle \sum }}_{k=1}^m\left[\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]>0\end{array}$, Eq. (9.32) should be satisfied.

${\delta }_{t,i,j}\le \frac{\varphi \left(A_i\right)-\varphi \left(A_j\right)}{{\displaystyle \sum }_{k=1}^m\left[\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]}$ (9.32)

(9.32)

While if ${\sum }_{k=1}^m\left[\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]<0$, Eq. (9.33) should be satisfied.

${\delta }_{t,i,j}\ge \frac{\varphi \left(A_i\right)-\varphi \left(A_j\right)}{{\displaystyle \sum }_{k=1}^m\left[\left(P_t\left(A_i,A_k\right)+P_t\left(A_k,A_j\right)\right)-\left(P_t\left(A_j,A_k\right)+P_t\left(A_k,A_i\right)\right)\right]}$ (9.33)