3.1 Entropy Weighting Methodology

Entropy is a concept introduced by thermodynamic, which had been used to measure the unavailability of a system’s energy to do work and also a measure of the disorder, namely, the confusion degree of the system; the higher the entropy the greater the disorder and vice versa.

Claude E. Shannon developed the concept of information entropy which is a measure of the uncertainty associated with a random variable (Zhang et al., 2011). The way to calculate entropy coefficient can avoid the deviation caused by some subjective reasons (Lotfi and Fallahnejad, 2010).

The entropy of indicator j can be calculated in Eq. (8.7):

$S_j=-k{\displaystyle \sum }_{i=1}^m{f}_{ij}\ln f_{ij}(j=1,2,\cdots ,n),$ (8.7)

(8.7)

where $f_{ij}$ and k can be calculated in Eqs. (8.8) and (8.9):

$f_{ij}=\frac{y_{ij}}{{\displaystyle \sum }_{i=1}^m{y}_{ij}},$ (8.8)

(8.8)

where $y_{ij}$ is the value of indicator j in alternative i, m represents the number of the alternatives, $f_{ij}$ indicates the ratio of indicator value of indicator j in sample i.

$k=\frac{1}{\ln m},$ (8.9)

(8.9)

where k represents entropy constant.

Eq. (8.8) can also normalize the raw data to eliminate anomalies with different measurement and scales because the process transform different scales and units among various indicators into common measurable units to allow for comparison of different indicator (Lotfi and Fallahnejad, 2010).

And assume $f_{ij}\ln f_{ij}=0$, when $f_{ij}=0$;

Then, the diversification of indicator j can be calculated in Eq. (8.10):

$D_j=1-S_j.$ (8.10)

(8.10)

The smaller the value of the entropy of indicator j, the bigger the effect of the indicator j is. Finally, the entropy weighting of indicator j can be computed by Eq. (8.11):

$W^{entropy}=\{{\omega }_j^{entropy}\}=\{D_j/\sum _{j=1}^n{D}_j\}(\mathrm{j}=1,2,\dots ,\mathrm{n}).$ (8.11)

(8.11)

3.2 Ideal Point Weighting Method

The principle of Ideal Point (IP) method is that the selected best alternative should have the shortest distance from the ideal point in geometrical sense (Ye et al., 2006; Hwang, 2013). The ideal point set and antiideal point set can be acquired by Eqs. (8.12) and (8.13):

$R^{+}=(y_1^{+},y_2^{+},\cdots ,y_n^{+}),$ (8.12)

(8.12)

$R^{-}=(y_1^{-},y_2^{-},\cdots ,y_n^{-}),$ (8.13)

(8.13)

where $y_j^{+}=\max (y_{ij}),y_j^{-}=\min (y_{ij}),i=1,2,\cdots ,m$.

The sum of Euclidean distance to the ideal point by weighting method can be calculated in Eq. (8.14):

$h_i^{+}=\sum _{j=1}^n{[w_j^{+}(y_{ij}-y_j^{+})]}^2=\sum _{j=1}^n{(w_j^{+})}^2{(y_{ij}-y_j^{+})}^{2}i=1,2,\cdots ,m.$ (8.14)

(8.14)

It can be supposed that the best weighting coefficient can minimize the sum of Euclidean distance to the ideal point, as shown in Eq. (8.15). The programming problem comprised by Eqs. (8.15)–(8.17) trends to make all the alternatives close to the ideal point, then the weighting coefficients can be obtained by Lagrange function, as shown in Eq. (8.18). The weighting coefficient vector can also be acquired as shown in Eq. (8.19).

$\min {\displaystyle \sum }_{i=1}^m{h}_i^{+}={\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{j=1}^n{(w_j^{*})}^2{(y_{ij}-y_j^{+})}^2,$ (8.15)

(8.15)

${\displaystyle \sum }_{j=1}^n{w}_j^{*}=1,$ (8.16)

(8.16)

$w_j^{*}>0,$ (8.17)

(8.17)

where $w_j^{*}$ is the weighting coefficient for indicator j determined by the first and second programming problem, respectively.

$w_j^{*}=\frac{\frac{1}{{\displaystyle \sum }_{i=1}^m{(y_{ij}-y_j^{+})}^2}}{{\displaystyle \sum }_{j=1}^n\frac{1}{{\displaystyle \sum }_{i=1}^m{(y_{ij}-y_j^{+})}^2}},$ (8.18)

(8.18)

$W^{*}={(w_1^{*},w_2^{*},\cdots ,w_n^{*})}^T.$ (8.19)

(8.19)

3.3 Delphi Methodology

Delphi method is a famous weighting methodology which has the characteristic of anonymity, feedback and convergence. The characteristic of anonymity means that the experts who participate in the forecast do not meet mutually, they fill the questionnaire (as shown in Table 8.1) of forecast by back to back, which makes the forecast be not interfered by the psychological factor, so as to reduce the one-sided views toward the authority and experts personally stubborn “the meeting syndrome.” It is advantageous to draw on the wisdom of the masses. The characteristic of Feedback means that it emphasizes on the communication and feedback of information. Each round of forecast would collect and collate every sort of opinions and materials of the preceding round. These opinions and materials would be delivered to experts along with the questionnaire in consultation time, which helps experts fully understand every kind of objective situation and points of view of other experts, so as to improve the comprehensiveness and reliability of forecast. And in these several rounds of the forecast consultation, various opinions of experts are compared, affect mutually and convince mutually, which helps each scattered opinions gradually centralize to the correct aspect to show convergence (Bin, 1998; Keeney et al., 2001; Van Zolingen and Klaassen, 2003) (Table 8.2).

The entire forecast procedures of Delphi method is shown in Fig. 8.1, the questionnaire about the importance of the indicator can be used to determine the score of each indicator, a fixed number of experts will be invited to fill in the questionnaire, there are five grades of importance, namely, lowly important (1 score), medium important (2 score, 3 score, and 4 score), and highly important (5 score), the importance of each indicator can be determined by the symbol (√) at the corresponding place by the expert, then the weighting coefficients of various indicators can be calculated in Eqs. (8.20) and (8.21):

${\overline{S}}_j=\frac{{\displaystyle \sum }_{p=1}^M{S}_p^j}{M},$ (8.20)

(8.20)

$W^{delphi}=\{{\omega }_j^{delphi}\}=\{\frac{{\overline{S}}_j}{{\displaystyle \sum }_{i=1}^N{\overline{S}}_j}\},$ (8.21)

(8.21)

where M represents the number of experts, N denotes the number of the indicators, $S_p^j$ is the score given by the expert p for indicator j, $W^{delphi}$ is the weighting coefficient vector, and ${\omega }_j^{delphi}$ is the weighting coefficient calculated by Delphi methodology.

3.4 Analytical Hierarchy Process

The AHP is a decision analysis method that considers both qualitative and quantitative information. The use of the AHP approach provided by Saaty (Saaty, 1980) to assess the criteria weightings in MCDM recently has become popular in different areas of system engineering (Su et al., 2010).

Suppose there are n criteria in some hierarchy, the pairwise comparison method proposed by Saaty (Saaty, 1980) can be used to establish the comparison matrix (denoted by matrix A), as shown in Eq. (8.22):

$A=\left[\begin{array}{cccc}1& a_{12}& \cdots & a_{1n}\\ a_{21}& 1& \cdots & a_{2n}\\ ⋮& \cdots & \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & 1\end{array}\right],$ (8.22)

(8.22)

where $a_{ij}$ denotes the relative importance of criteria i comparing with j.

The relative importance of criteria j comparing to i can calculated by Eq. (8.23):

$a_{ji}=\frac{1}{a_{ij}},a_{ij}>0,i,j=1,2,\cdots ,n.$ (8.23)

(8.23)

As for different systems and different implementers, the result of comparison matrix A is not absolutely the same with each other. With the comparison matrix, the weighting coefficients of each indicator can be acquired by calculating the principal eigenvector of the comparison matrix, as shown in Eq. (8.24):

$\left[\begin{array}{cccc}1& a_{12}& \cdots & a_{1n}\\ a_{21}& 1& \cdots & a_{2n}\\ ⋮& \cdots & \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & 1\end{array}\right]\left|\begin{array}{c}{w}_1\\ w_1\\ ⋮\\ w_n\end{array}\right|={\lambda }_{\max }\left|\begin{array}{c}{w}_1\\ w_1\\ ⋮\\ w_n\end{array}\right|,$ (8.24)

(8.24)

where ${(w_1,w_2,\cdots ,w_n)}^T$ is the maximal eigenvector of matrix A and ${\lambda }_{\max }$ is the maximal eigenvalue of matrix A. The maximal eigenvector and maximal eigenvalue can be calculated in the form as shown in Eq. (8.25) by the tool box of MATLAB when the comparison matrix A have been determined.

$[W_{\max },{\lambda }_{\max }]=eig(A),$ (8.25)

(8.25)

where ${\lambda }_{\max }$ and $W_{\max }$ are the maximal eigenvalue and maximal eigenvector of the comparison matrix, respectively.

Then weighting coefficient vector can be calculated by normalization of the maximal eigenvector, as shown in Eq. (8.26):

$W={(\frac{w_1}{{\displaystyle \sum }_{i=1}^n{w}_i},\frac{w_2}{{\displaystyle \sum }_{i=1}^n{w}_i},\cdots ,\frac{w_n}{{\displaystyle \sum }_{i=1}^n{w}_i})}^T,$ (8.26)

(8.26)

where $W$ is the weight coefficient vector, $w_i$ represents the weight of indicator i, n represents the total number of the indicators.

If $a_{ik}=a_{ij}{a}_{jk},i,j,k=1,2,\cdots ,n$, then the comparison matrix A can be recognized as consistent matrix. Theory proves that if n-dimensional comparison matrix is a consistent matrix, its maximal eigenvalue must be n. But, it is difficult to establish a comparison matrix which is consistent matrix absolutely. In actual cases, some comparison matrix that meets the condition of consistency check can be recognized as consistent matrix. Consistency ratio is the common method to judge whether a comparison matrix is consistent or not, as shown in Eq. (8.27):

$CR=\frac{CI}{RI},$ (8.27)

(8.27)

where CR is consistency ratio, CI is consistency index, RI is the average random index with the same dimension with A.

The value of average random index can be acquired in Table 8.3, the consistency index can be computed in Eq. (8.28):

$CI=\frac{{\lambda }_{\max }-n}{n-1},$ (8.28)

(8.28)

where ${\lambda }_{\max }$ represents the maximal eigenvalue of the comparison matrix A and n represents the dimension of the matrix.

When CR < 0.1, the matrix can be acceptable as consistent matrix, contrarily CR ≥ 0.1, the matrix should be modified until an acceptable one.

4.1 TOSPIS Ranking Method

TOPSIS which had been developed by Hwang and Yoon in 1980s holds the view that the best alternative should have the shortest distance from the ideal point and the longest distance from the antiideal point (Ching-Lai and Yoon, 1981). The optimal solution is the one near the ideal point and distant from the antiideal point (Chu and Su, 2012). But it has been found that sometime there is a solution that has the minimum Euclidean distance from the ideal solution may also has a shortest distance from the antiideal solution as compared to other alternatives, TOPSIS tries to find the solutions that are simultaneously close to the ideal point and far from the antiideal point (Shanian and Savadogo, 2006).

The ideal point comprises all the best values attainable from the criteria, whereas the antiideal point consists of all worst values attainable from the criteria (Salmeron et al., 2012). The ideal point and antiideal point can be acquired by Eqs. (8.6) and (8.7), and if the sample approaches the ideal point and is far away from antiideal point, the sample will be more superior; Minkowski distance methodology can be used to measure the distance from sample i to the ideal point and antiideal point, as shown in Eqs. (8.29) and (8.30):

$D_i^{+}={\{{\displaystyle \sum }_{j=1}^n{\omega }_j{(y_{ij}-Y_j^{+})}^p\}}^{1/p},$ (8.29)

(8.29)

$D_i^{-}=\{{{\displaystyle \sum }_{j=1}^n{\omega }_j{(y_{ij}-Y_j^{-})}^p\}}^{1/p},$ (8.30)

(8.30)

where $D_i^{+}$ represents the distance from sample i to the ideal point, $D_i^{-}$ represents the distance from sample i to the antiideal point, ${\omega }_j$ represents the weighting coefficient of index j, p represents exponential coefficient taking value 2 in this paper.

The closeness coefficient is defined in Eq. (8.31). It indicates the closeness level of the sample to the ideal point and the farness level of the sample to the antiideal point; so, the bigger the first coefficient is, the more superior the sample will be.

$C^i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}},$ (8.31)

(8.31)

where $C^i$ represents the closeness coefficient of the i-th option.

In other words, the higher the closeness coefficient means the better the option. Thus, the closeness coefficients can be used to measure the superiority of the alternatives (Fig. 8.2).

The procedure of TOPSIS methodology can be found in Fig. 8.3, after the closeness coefficients have been calculated, the superiority of the scenarios can be determined.

4.2 Data Envelopment Analysis

The DEA was developed by Charnes and Cooper in 1978 (Charnes et al., 1978), which had been widely used to assessment alternatives with the inputs and outputs of these systems. This method is to define the efficiency measure by assign to each unit the most favorable weights, and it has two significant advantages, the first is that the weighting sets in different units are different, the second is that if a unit turns out to be inefficient compared to the other ones when the most favorable weights are chosen, it proves that it does not depend on the choice of the weights (Basso and Funari, 2001).

DEA can be used to measure the relative efficiencies of different decision-making units that use similar inputs to produce similar outputs where the multiple inputs and outputs are incommensurate in nature (Li and Reeves, 1999). Each alternative can be considered as a system which also has been called decision-making unit(DMU), as shown in Fig. 8.4.

The meaning of the symbols in Fig. 8.4 have been defined as follows:

The efficiency of a DMU j can be formulated by the ratio of weighted sum of outputs to weighted sum of inputs (Basso and Funari, 2001), as shown in Eq. (8.32):

$h_j=\frac{{\displaystyle \sum }_{i=1}^p{v}_i{y}_{ij}}{{\displaystyle \sum }_{r=1}^m{u}_r{x}_{rj}}$ (8.32)

(8.32)

The DEA efficiency of the target system j0 can be calculated by solving the programming problem comprised by Eqs. (8.33)–(8.36):

$\max h_0=\frac{{\displaystyle \sum }_{i=1}^p{v}_i{y}_{ij0}}{{\displaystyle \sum }_{r=1}^m{u}_r{x}_{rj0}},$ (8.33)

(8.33)

Subject to

$\frac{{\displaystyle \sum }_{i=1}^p{v}_i{y}_{ij}}{{\displaystyle \sum }_{r=1}^m{u}_r{x}_{rj}}\le 1(j=1,2,\cdots ,t)$ (8.34)

(8.34)

$u_r\ge \epsilon r=1,2,\cdots ,m$ (8.35)

(8.35)

$v_i\ge \epsilon i=1,2,\cdots ,p$ (8.36)

(8.36)

where $\epsilon$ is a nonarchimedean construct.

The constraints [inequality (8.34)] indicate that the upper bound of the efficiency of the DMU is 100%, namely, the efficiency cannot exceed 1.

The model can be transformed into matrix form, as shown in Eqs. (8.37)–(8.40):

$\max h_0=\frac{v^T{y}_0}{u^T{x}_0},$ (8.37)

(8.37)

$\frac{v^T{y}_j}{u^T{x}_j}\le 1,$ (8.38)

(8.38)

$u\ge \epsilon ,$ (8.39)

(8.39)

$v\ge \epsilon ,$ (8.40)

(8.40)

where

$u={(u_1,u_2,\cdots ,u_m)}^T,v={(v_1,v_2,\cdots ,v_p)}^T,x_j={(x_{1j},x_{2j},\cdots ,x_{mj})}^T,y_j={(y_{1j},y_{2j},\cdots ,y_{pj})}^T,x_0={(x_{1j0},x_{2j0},\cdots ,x_{mj0})}^T,y_0={(y_{1j0},y_{2j0},\cdots ,y_{pj0})}^T,$

with Charnes–Cooper transformation, as shown in Eq. (8.41):

$\{\begin{array}{ll}t\hfill & =1/u^T{x}_0\hfill \\ \mu \hfill & =tu\hfill \\ \omega \hfill & =tv\hfill \end{array}.$ (8.41)

(8.41)

Then, the equivalent linear programming can be acquired, as shown in Eqs. (8.42)–(8.46):

$\max {\omega }^T{y}_0,$ (8.42)

(8.42)

Subject to

${\displaystyle \sum }_{r=1}^m{\mu }_r{x}_{rj0}=1$ (8.43)

(8.43)

${\displaystyle \sum }_{i=1}^p{\omega }_i{y}_{ij}-{\displaystyle \sum }_{r=1}^m{\mu }_r{x}_{rj}\le 0(j=1,2,\cdots ,t)$ (8.44)

(8.44)

$u\ge \epsilon$ (8.45)

(8.45)

$v\ge \epsilon$ (8.46)

(8.46)

Subsequently, the linear programming problem can be transformed into the following form, as shown in Eqs. (8.47)–(8.51):

$\max ({\mu }^T,{\omega }^T)\left(\begin{array}{c}0\\ y_0\end{array}\right)$ (8.47)

(8.47)

${\omega }^T{y}_j-{\mu }^T{x}_j\le 0(j=1,2,\cdots ,t)$ (8.48)

(8.48)

$u\ge \epsilon$ (8.49)

(8.49)

$v\ge \epsilon$ (8.50)

(8.50)

${\mu }^T{x}_{j0}=1$ (8.51)

(8.51)

According to the duality theory of linear programming, it can be transformed into the following form comprised by Eqs. (8.52)–(8.57):

$\min \begin{array}{cc}\theta -\epsilon ({\displaystyle \sum }_{r=1}^m{s}_r^{+}+{\displaystyle \sum }_{i=1}^p{s}_i^{-})& \end{array}$ (8.52)

(8.52)

Subject to

${\displaystyle \sum }_{j=1}^t{x}_{rj}{\lambda }_j+s_r^{-}-\theta x_{rjo}=0$ (8.53)

(8.53)

${\displaystyle \sum }_{j=1}^t{y}_{ij}{\lambda }_j-s_i^{+}-y_{ijo}=0$ (8.54)

(8.54)

${\lambda }_j\ge 0(j=1,2,\dots ,t)$ (8.55)

(8.55)

$s_r^{-}\ge 0(r=1,2,\dots ,m)$ (8.56)

(8.56)

$s_i^{+}\ge 0(i=1,2,\dots ,p)$ (8.57)

(8.57)

The assessment systems can be judged based on the following two definitions (Ye et al., 2006):

4.3 Preference Ranking Organization Method for Enrichment Evaluation

PROMETHEE that had been developed in 1980s, can use the outranking principle to rank the alternatives (Brans and Vincke, 1985). The PROMETHEE method has been extended to 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 criteria are intervals) and PROMETHEE GAIA (graphical representation of PROMETHEE) (Zhang et al., 2009). The PROMETHEE method can handle data that are known with a reasonable degree of accuracy and have fixed numerical values, and it is a ranking method quite simple in conception and application compared to other multicriteria making methods (Goumas and Lygerou, 2000; Li and Li, 2010). Consequently, PROMETHEE methods have taken an important place among the existing outranking multiple criteria methods (De Keyser and Peeters, 1996).

In PROMETHEE II a complete pre-order (complete ranking) of alternatives is obtained from the net flow that calculated from each alternative. One of the advantages of PROMETHEE over other outranking methods, such as ELECTRE methods, is related to the fact that the decision-makers find it easy to understand the concepts and parameters inherent in the method, which makes the preference modeling simpler and, consequently, increases the effectiveness of applying the methods (Silva et al., 2010). The analysis procedures using the PROMETHEE includes: (a) Establish an alternatives and criterion matrix; (b) selecting a preference function; and (c) calculating the preference index (Chou et al., 2007).

A preference function, as defined in PROMETHEE methods, is turning a difference between two alternatives on a criterion into a value between 0 and 1. The accuracy of the evaluation depends much on the selection of the preference; there are six main types of preference function, which cover most of the practical situations (Ye et al., 2006; Chou et al., 2007). The proposed six types of generalized criteria have been shown in Table 8.4.

Table 8.4

The Shape of the Six Possible Types of Generalized Criteria

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

Gaussian type has the advantage of the sensitive to small variations of the PROMETHEE II input parameters (Parreiras and Vasconcelos, 2007), and it also contains continuity (Chou et al., 2007). Consequently, Gaussian type has been chosen for evaluation in this paper.

Multicriteria preference index for a pair of alternatives $A_i$ and $A_k$ has been defined as follows:

$H(A_i,A_k)={\displaystyle \sum }_{j=1}^n{w}_j{P}_j(A_i,A_k),$ (8.58)

(8.58)

where $P_j(A_i,A_k)$ represents the preference function; $w_j$ represents the weighting coefficient of the indicator j.

The positive flow can be calculated with Eq. (8.59):

${\varphi }^{+}(A_i)={\displaystyle \sum }_{k=1}^{m}H(A_i,A_k)(i=1,2,\cdots ,m).$ (8.59)

(8.59)

The negative flow can be calculated with Eq. (8.60):

${\varphi }^{-}(A_i)={\displaystyle \sum }_{k=1}^{m}H(A_k,A_i)(i=1,2,\cdots ,m).$ (8.60)

(8.60)

Then the net flow can be calculated with Eq. (8.61):

$\varphi (A_i)={\varphi }^{+}(A_i)-{\varphi }^{-}(A_i)(i=1,2,\cdots ,m).$ (8.61)

(8.61)

In PROMETHEE II, the net flow can be obtained from the difference between the positive and negative flow and it give an idea of how much each alternative is preferred to the others. Hence, its value can be used to rank all alternatives, in such way that higher values of the net flow correspond to better solutions (Albadvi et al., 2007; Parreiras and Vasconcelos, 2007). The standards have been shown in Eqs. (8.62) and (8.63):

$if\varphi (A_i)\succ \varphi (A_k)then{A}_i\succ A_K,$ (8.62)

(8.62)

$if\varphi (A_i)~\varphi (A_k)then{A}_i~A_K,$ (8.63)

(8.63)

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

4.4 Principal Component Analysis

PCA is a mathematical tool which performs a reduction in data dimensionality and allows the visualization of underlying structure in experiment data and relationships between data and samples, it is a multivariate statistical techniques used to identify important components or factors that explain most of the variances of a system. The technique has the ability to reduce the number of variables to a small number of indices (i.e., principal components or factors) while attempting to preserve the relationships present in the original data (Patras et al., 2011).

Mathematically, PCA normally involves the following five major steps: (1) Start by coding the variables to have zero means and unit variance, i.e., standardization of the measurement to ensure that they all have equal weights in the analysis; (2) calculate the covariance matrix; (3) find the eigenvalue and the corresponding eigenvectors; and (4) discard any components that only account for a small proportion of the variation in data sets. The factor correlation coefficient that is greater than 0.75 (or 75%) was considered significant (Ouyang, 2005). The procedure of principal component analysis evaluation has been described as follows:

Step 1: Collect the data about the characteristics (criteria) of the samples, let the original decision-making matrix.

$X=\left|\begin{array}{cccc}{x}_{11}& x_{12}& \cdots & x_{1n}\\ x_{21}& x_{22}& \cdots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{m1}& x_{m2}& \cdots & x_{mn}\end{array}\right|=(X_1,X_2,\cdots ,X_n),$ (8.64)

(8.64)

where m is the number of the sample and n is the number of the characteristic; x_ij represents the value of the j(th) characteristic of the i(th) sample.

Step 2: Transform all the criteria in the original decision-making matrix to benefit type, the transformation can be carried out according to the type of the characteristic (criteria).

$t_{ij}=\{\begin{array}{l}\frac{x_{ij}-\underset{i=1,2,\cdots ,m}{\min }\left\{x_{ij}\right\}}{\underset{i=1,2,\cdots ,m}{\max }\left\{x_{ij}\right\}-\underset{i=1,2,\cdots ,m}{\min }\left\{x_{ij}\right\}}\begin{array}{c},\mathrm{the}j(\text{th})\mathrm{characteristic}\mathrm{is}\mathrm{the}\mathrm{benefit}\mathrm{type}\end{array}\\ \frac{\underset{i=1,2,\cdots ,m}{\max }\left\{x_{ij}\right\}-x_{ij}}{\underset{i=1,2,\cdots ,m}{\max }\left\{x_{ij}\right\}-\underset{i=1,2,\cdots ,m}{\min }\left\{x_{ij}\right\}}\begin{array}{c},\mathrm{the}j(\text{th})\mathrm{characteristic}\mathrm{is}\mathrm{the}\mathrm{cost}\mathrm{type}\end{array}\end{array},$ (8.65)

(8.65)

where benefit criteria is the-larger-the-better type and cost-criteria is the-smaller-the better.

Step 3: Standard transformation.

$y_{ij}=\frac{t_{ij}-{\displaystyle {\displaystyle \sum }_i^m}{t}_{ij}/m}{\sqrt{{\displaystyle \sum }_{i=1}^m{(t_{ij}-{\displaystyle \sum }_i^m{t}_{ij}/m)}^2/m-1}}$ (8.66)

(8.66)

Step 4: Calculate the correlation coefficient matrix. The element of correlation coefficient matrix can be calculated by Eq. (8.67):

$r_{ij}=\left(\frac{Cov(y_{si},y_{sj})}{\sigma (y_{si})\times \sigma (y_{sj})}\right)\begin{array}{c},i=1,2,\cdots ,n\end{array};j=1,2,\cdots ,n$ (8.67)

(8.67)

$R={\left\{r_{ij}\right\}}_{m\times n}\begin{array}{c},i=1,2,\cdots ,n\end{array};j=1,2,\cdots ,n$ (8.68)

(8.68)

where $Cov(y_{si},y_{sj})$ is the covariance of sequences $y_{si}$ and $y_{sj}$;$\sigma (y_{si})$ is the standard deviation of sequences $y_{si}$,$\sigma (y_{sj})$ is the standard deviation of sequences $y_{sj}$.

Step 5: Solve the eigenvalue and eigenvector, then calculate the contribution rate H and the cumulative contribution rate TH. The eigenvalues can be determined by Eq. (8.69), the contribution rate and cumulative contribution rate can be calculated by Eqs. (8.70) and (8.71), respectively.

$(R-{\lambda }_k{I}_n)V_k=0,$ (8.69)

(8.69)

where λ_k represents the eigenvalue, I_n represents the unit matrix of n-order, V_k is the corresponding eigenvector of λ_k.

$H_k=\frac{{\lambda }_k}{{\displaystyle {\displaystyle \sum }_{K=1}^n}{\lambda }_k}$ (8.70)

(8.70)

$T{H}_l=\frac{{\displaystyle {\displaystyle \sum }_{k=1}^l}{\lambda }_k}{{\displaystyle {\displaystyle \sum }_{k=1}^n}{\lambda }_k}$ (8.71)

(8.71)

Step 6: Express the principal component. Select the first t principal component to make the cumulative contribution rate is greater than 85%, then it can be recognized as significant, it means that the original n characteristics can be expressed by the new t principal components. Then the first t principal component can be determined by Eq. (8.72):

$\left|\begin{array}{c}{P}_1\\ P_2\\ ⋮\\ P_t\end{array}\right|=\left|\begin{array}{c}{V}_1\\ V_2\\ ⋮\\ V_t\end{array}\right|\left|\begin{array}{c}{C}_1\\ C_2\\ ⋮\\ C_n\end{array}\right|,$ (8.72)

(8.72)

where P_s represents the s(th) principal component, and V_k is the k(th) eigenvector which has n elements, as shown in Eq. (8.73):

$V_k=\left(v_{k1},v_{k1},\cdots ,v_{kn}\right)$ (8.73)

(8.73)

Step 7: Calculate the weight of the principal component.

${\omega }_k=\frac{H_k}{{\displaystyle \sum }_{k=1}^t{H}_k}\begin{array}{c},k=1,2,\cdots ,t\end{array}$ (8.74)

(8.74)

where ω_k is the weight of the k(th) principal, H_k represents the contribution of the k(th) eigenvale.

Step 8: Determine the evaluation function of each sample:

$\left|\begin{array}{c}{F}_1\\ F_2\\ ⋮\\ F_m\end{array}\right|=\left({\omega }_1,{\omega }_2,\cdots ,{\omega }_t\right)\left|\begin{array}{c}{P}_1\\ P_2\\ ⋮\\ P_t\end{array}\right|.$ (8.75)

(8.75)

Step 9: Rank the sequence of the samples according to the rule that the larger the score of the evaluation function the better the sample.

5.1 Weighting Coefficient Calculation

The hierarchy for AHP study has been shown in Fig. 8.5. AHP has been used to calculate the weighting coefficients of the evaluation indicators; the comparison matrix of the four aspects (performance, market, environment, and society) has been show in Eq. (8.76). The principal eigenvalue and the consistency have been calculated, as shown in Eqs. (8.77) and (8.78). It can be seen that if the comparison matrix satisfies the consistency check, then the weighting coefficients can be calculated, as shown in Table 8.7. The comparison matrixes of the indicators in each aspect have been shown in Eqs. (8.79)–(8.82); likewise, with the consistency check, the corresponding weighting coefficients of the indicators in each aspect and the final indicator of each criteria can be calculated, as shown in Table 8.8.

$\left|\begin{array}{ccccc}& Performance& Market& Environment& Society\\ Performance& 1& 3& 1& 2\\ Market& 1/3& 1& 1/5& 1\\ Environment& 2& 4& 1& 3\\ Society& 1/4& 1& 1/3& 1\end{array}\right|$ (8.76)

(8.76)

${\lambda }_{\max }=4.0365$ (8.77)

(8.77)

$CR=0.0135<0.1$ (8.78)

(8.78)

$\left|\begin{array}{ccccc}& Ef& EC& CC& Li\\ Ef& 1& 4& 1& 5\\ EC& 1/4& 1& 1/2& 2\\ CC& 1& 2& 1& 2\\ Li& 1/5& 1/3& 1/2& 1\end{array}\right|$ (8.79)

(8.79)

$\left|\begin{array}{cccc}& N{O}_x& C{O}_2& KI\\ N{O}_x& 1& 1/2& 1/3\\ C{O}_2& 2& 1& 1/2\\ KI& 3& 2& 1\end{array}\right|$ (8.80)

(8.80)

$\left|\begin{array}{ccc}& EM& WM\\ EM& 1& 1\\ WM& 1& 1\end{array}\right|$ (8.81)

(8.81)

$\left|\begin{array}{ccc}& A& NJ\\ A& 1& 1\\ NJ& 1& 1\end{array}\right|$ (8.82)

(8.82)

Ten experts had been invited to fill the questionnaire, all of them are researchers on sustainable energy and sustainability, the result of the survey has been shown in Table 8.9, then the weighting coefficients of the indicators can be calculated by Delphi methodology, as shown in Table 8.10.

The weighting coefficients of the indicators calculated by the objective methodologies (entropy and ideal point), and the average of the weighting coefficients calculated by the four methodologies have also been shown in Table 8.10. It can be seen that the weighting coefficients of the indicators that have been calculated by different methodologies are different, even the results calculated by the two objective are also different. The objective methodology for the calculation of weighting coefficients can reflect the essence of the things and the subjective can reflect the preference of the decision-makers.

5.2 The Results of MCDM

The closeness indexes of the five energy scenarios by TOPSIS under different weighting coefficients have been show in Table 8.11, and the ranks of the energy scenarios by TOPSIS with different weighting coefficients has been show in Fig. 8.6.

It is obvious that different weighting coefficients using in TOPSIS may cause different ranks, but the results are relatively stable in this case, wind and gas turbine has been recognized as the most sustainable, and follows by Photovoltaic, PAFC and SOFC (from the best to the worst).

Wind and gas turbine have been recognized as the most excellent energy scenarios, the result is consistent with the original research, in study proposed by Afgan and Carvalho (2004), different weighting sets have been used in the synthesizing function, and wind and gas turbine are also obtained high priority.

The implementation of PROMETHEE requires two additional types of information, namely: (a) Information on the relative importance that is the weights of the criteria considered and (b) information on the decision-makers preference function, which he/she uses when comparing the contribution of the alternatives in terms of each separate criterion (Dağdeviren, 2008).

The weighting coefficients were determined by Entropy, IP, AHP, and Delphi; average has been used in PROTHEE II to evaluate the five energy scenarios. The outgoing flow using PROTHEE II under different weighting coefficients has been shown in Table 8.12, and the ranks of the scenarios with different weighting coefficients have been shown in Fig. 8.6.

The results are very similar with that determined by TOPSIS, and also fit well with original research. Wind and gas turbine has also been recognized as the most sustainable energy scenarios; SOFC has been recognized as the worst in most of the situations, but the ranks are also not all the same. It can also be concluded that different weighting weightings may also cause different ranks in PROTHEE methodology.

In the DEA, five decision-making units (DMUs) have been considered, EC, CC, NOx, CO₂, KI, A that are negative indicators have been used as the inputs, and Ef, Li, EM, WM and NJ that are positive indicators have been used as the outputs in the DMU.

The DEA efficiencies and rank of the five energy scenarios have been shown in Table 8.13, the energy scenarios except SOFC have all been recognized as efficient unit, consequently, SOFC has been recognized as the worst energy scenario whereas the superiority of the other four scenarios cannot be differed. In the original research, SOFC has also obtained a very low rating in most of situations that have been carried. It proves that DEA has the ability to find out the relative efficient and inefficient scenarios, but it cannot identify the best scenarios from the efficient ones.

Table 8.13

DEA Efficiencies and Ranking of the Five Energy Scenarios

Option	$s_1^{-}$	$s_2^{-}$	$s_3^{-}$	$s_4^{-}$	$s_5^{-}$	$s_6^{-}$	$s_1^{+}$	$s_2^{+}$	$s_3^{+}$	$s_4^{+}$	$s_5^{+}$	$\theta$	Ranking
PAFC	0	0	0	0	0	0	0	0	0	0	0	1	1
SOFC	0.0818	2475.92	0	1.2510	0	1.2900	0	4.8584	39.4275	105.9292	2.1227	0.8339	2
Gas Turbine	0	0	0	0	0	0	0	0	0	0	0	1	1
Photovoltaic	0	0	0	0	0	0	0	0	0	0	0	1	1
Wind	0	0	0	0	0	0	0	0	0	0	0	1	1

This study applied principal component analysis to evaluate the integrated performance of the energy scenarios; Table 8.14 shows a summary of the coefficients of the three principal components and the relevant statistics from the PCA. The eigenvalue is a measure of the variance accounted for by the corresponding principal component. The first and largest eigenvalue accounts for most of the variance, and the second the second largest amounts of variance, and so on. Principal component can be ranked according to their ability to explain variance in the original data set (Lam et al., 2008).

It can be seen that the first three PCs explain 96.9834% of the original data variance for the systems analysis. In the first PC, (PC1) is contributed mainly by the criteria lifetime, European market, world market, number of jobs, and Tokyo index; the second PC2 is heavily loaded by the contribution from efficiency, electricity cost, CO₂, and area; and the main constitution of PC3 is nearly the same with PC2.

The three principal components can be calculated as linear combinations of the original 11 indicators, the formulas for calculated the three PCs have been show in Eq. (8.83).

$\left|\begin{array}{c}PC1\\ PC2\\ PC3\end{array}\right|=\left|\begin{array}{ccccccccccc}-0.0713& 0.3282& 0.3815& 0.3880& 0.3808& 0.2432& 0.2538& -0.3431& 0.1294& -0.3814& 0.2063\\ -0.4362& 0.3094& -0.0350& -0.0581& -0.0335& 0.4215& -0.2936& 0.1828& 0.4643& 0.0504& -0.4392\\ -0.4049& -0.1436& -0.2506& 0.1915& 0.2600& -0.2449& -0.3742& -0.3470& -0.4101& -0.2457& -0.3152\end{array}\right|\times \left|\begin{array}{c}Ef\\ Li\\ EM\\ WM\\ NJ\\ EC\\ CC\\ N{O}_X\\ C{O}_2\\ KI\\ A\end{array}\right|$ (8.83)

(8.83)

According to the corresponding variances of the three PCs, the value function of each energy scenario can be acquired by weighting the three PCs, as shown in Eq. (8.84).

$Score=0.5822PC1+0.2893PC2+0.1285PC3$ (8.84)

(8.84)

Then, the energy scenarios can be expressed by the three principal components as shown in Table 8.15, and the final score of each energy scenario can be acquired, as shown in Table 8.16, the ranks of the energy scenarios has been shown in Fig. 8.7.

In the PCA, gas turbine has been recognized as the best scenario; wind has been recognized as the second best scenario; photovoltaic, PAFC, and SOFC have been assign at the third, fourth and fifth place, respectively. The result calculated by PCA is basically in accordance with that calculated by TOPSIS, PROTHEE, and DEA; gas turbine and wind have been recognized as the most excellent, whereas SOFC has been recognized as the worst. The result is also consistent with the original research.