4.2.1.1. Quantifying the Qualitative Criteria

In decision-making problems, some of the criteria such as OPL, maturity, UFCR, IEL, and LAEM are qualitative and it is not easy to quantify them. The importance of a qualitative criterion depends on the technical data, and the opinion and knowledge of the decision-maker. Therefore, fuzzy-MCDM must propose a method to quantify qualities such as “very bad,” “bad,” “middle,” “good,” and “very good.” In the fuzzy-MCDM, a linguistic judgment is used and the corresponding triangular fuzzy numbers proposed by [4] are used to quantify the qualitative criteria. The linguistic judgments and their corresponding fuzzy numbers are presented in Table 4.1 [5, 6].

In what follows the uppercase letters show the fuzzy numbers and the lowercase letters indicate the real numbers.

4.2.1.2. Generating the Judgment Matrix

Fuzzy decision-making for the basic components of a CCHP system starts by generating a judgment matrix as follows:

$J={\left[J_{ij}\right]}_{m\times n}$

(4-5)

where J_ij is our judgment about the ith option based on the jth criteria, with i = 1, 2…, m and j = 1, 2…, n. In addition, m and n are the number of alternatives for each component and the number of criteria, respectively.

4.2.1.3. Normalization of the Judgment Matrix

To allow criteria to be properly evaluated while comparing options, all the criteria must fall in the same range. For this purpose, the judgment matrix is normalized.

If the criterion is “the higher the better” such as the FESR then

$J_{ij}^N=\frac{J_{ij}}{J_j^{\uparrow }}$

(4-6)

If the criterion is “the lower the better” such as the investment cost then

$J_{ij}^N=\frac{J_j^{\downarrow }}{J_{ij}}$

(4-7)

where in the preceding equations

$\begin{array}{l}{J}_j^{\uparrow }=\max \{J_{ij}\left|i=1,2\dots ,m\}\right.\\ J_j^{\downarrow }=\min \{J_{ij}\left|i=1,2\dots ,m\}\right.\end{array}$

(4-8)

If in the normalization process a component of the triangular fuzzy number becomes greater than 1, it will be substituted with 1. Therefore, the normalized judgment matrix (J^N) would be as follows:

$J^N={\left[J_{ij}^N\right]}_{m\times n}$

(4-9)

4.2.1.4. Calculating the Normalized Weight Matrix

In MCDM methods, the degree of importance of each criterion with respect to the other criteria is considered by calculating the weight of each criterion. For this purpose, the pairwise comparison (PC) matrix between the n criteria is made as follows:

$PC={\left[P{C}_{ij}\right]}_{n\times n}$

(4-10)

where PC_ij represents the importance of the ith criterion with respect to the jth criterion. In the fuzzy-MCDM, a linguistic pairwise comparison presented by [4] is used to provide the PC matrix as in Table 4.2.

The equation to calculate the weight of each criterion or subcriterion in the fuzzy eigenvector of the PC matrix is

$E_i={\left[\prod _{j=1}^{n}P{C}_{ij}\right]}^{1/n}$

(4-11)

Finally, the normalized weight matrix for each criterion and subcriterion would be calculated as follows:

$W_i=\frac{E_i}{\sum _{i=1}^n{E}_i}$

(4-12)

4.2.1.5. Finding the Ideal and Anti-ideal Solutions for All Criteria

The case in which all the subcriteria are the best simultaneously is the ideal solution (I^↑) and the case in which all the criteria are the worst simultaneously would be the anti-ideal solution (I^↓). Therefore, the best and the worst solutions would be as follows:

$\begin{array}{l}{I}^{\uparrow }=\left[I_1^{\uparrow },\mathrm{\hspace{0.17em} }{I}_2^{\uparrow }\mathrm{\hspace{0.17em} }\dots ,\mathrm{\hspace{0.17em} }{I}_n^{\uparrow }\right]\\ I_j^{\uparrow }=\max \{\left.J_{ij}^N\right|\mathrm{\hspace{0.17em} }i=1,\mathrm{\hspace{0.17em} }2,\mathrm{\hspace{0.17em} }\dots ,\mathrm{\hspace{0.17em} }m\}\end{array}$

(4-13)

$\begin{array}{l}{I}^{\downarrow }=\left[I_1^{\downarrow },I_2^{\downarrow }\dots ,I_n^{\downarrow }\right]\\ I_j^{\downarrow }=\min \{\left.J_{ij}^N\right|i=1,2,\dots ,m\}\end{array}$

(4-14)

4.2.1.6. Finding the Weighted Distance from the Ideal and Anti-ideal Solutions

In this step, we calculate the weighted distance. The distance from the ideal and anti-ideal solutions is calculated as follows:

${d_{ij}}^{\uparrow \downarrow }(I_j^{\uparrow \downarrow },J_{ij}^N)=\sqrt{\frac{1}{4}({(a_j^{\uparrow \downarrow }-a_{ij}^N)}^2+2{(b_j^{\uparrow \downarrow }-b_{ij}^N)}^2+{(c_j^{\uparrow \downarrow }-c_{ij}^N)}^2)}$

(4-15)

The weighted distance can be calculated according to the following equation:

$D_i^{\uparrow \downarrow }=\sum _{j=1}^n{\beta }_j{W}_j{d}_{ij}^{\uparrow \downarrow },i=\mathrm{1,2},\dots ,m$

(4-16)

where β ∈ {0, 1} is the criteria coefficient and is used for the purpose of single- or multicriteria decision-making. When a criterion is supposed to be counted in the decision-making process β = 1, otherwise β = 0.

4.2.1.7. Calculating the Closeness Number (CN)

The closeness of different options to the ideal solution is calculated according to the following equations:

$\begin{array}{l}N{D}^{\downarrow }=\min \{D_i^{\downarrow }i=1,2,\dots ,m\},N{D}^{\uparrow }=\max \{D_i^{\downarrow }i=1,2,\dots ,m\}\\ P{D}^{\downarrow }=\min \{D_i^{\uparrow }i=1,2,\dots ,m\},P{D}^{\uparrow }=\max \{D_i^{\uparrow }i=1,2,\dots ,m\}\end{array}$

(4-17)

${\alpha }_i^{\uparrow \downarrow }=d(D_i^{\uparrow \downarrow },P{D}^{\downarrow \uparrow })+d(D_i^{\downarrow \uparrow },N{D}^{\uparrow \downarrow })$

(4-18)

Based on this, the closeness number (CN_i) is used in the final decision-making process and is calculated as follows:

$C{N}_i=\frac{{\alpha }_i^{\downarrow }}{{\alpha }_i^{\downarrow }+{\alpha }_i^{\uparrow }},i=1,2,\dots ,m$

(4-19)

It is obvious that 0 ≤ CN_i ≤ 1, therefore as CN_i approaches 1 it is getting closer to the ideal solution; on the contrary if CN_i approaches zero it is getting closer to the anti-ideal solution.

4.2.2.1. Quantifying the Qualitative Criteria

In order to quantify the qualitative criteria, in the GIA, a letter grade from A to E is utilized, in which A represents the very good and E represents the very bad. Moreover, the scale shown in Figure 4.4 is used to quantify these letter judgments [7-9].

4.2.2.2. Generating the Judgment Matrix

The grey-MCDM starts by writing the judgment matrix as follows:

$J={⌊J_{ij}⌋}_{m\times n}$

(4-20)

where J_ij is our judgment about the ith option based on the jth criteria with i = 1, 2…, m and j = 1, 2…, n. In addition, m and n are the number of alternatives for each component and the number of criteria, respectively.

For example, if the problem is to find the best type of prime mover, and there are 5 types of prime movers and 15 subcriteria, m = 5 and n = 15. If the third type of prime mover is an MGT and the first subcriteria is the FESR, then J₃₁ = FESR_MGT.

4.2.2.3. Normalization of the Judgment Matrix

In order to normalize the judgment matrix from the previous section, if the criterion (J_ij) is a profit for the project and is “the higher the better” such as the UFCR, then it is normalized as below:

$J_{ij}^N=\frac{J_{ij}-J_j^{\downarrow }}{J_j^{\uparrow }-J_j^{\downarrow }}$

(4-21)

If the J_ij is a cost for the project and is “the lower the better” such as I_OM, then it is normalized as follows:

$J_{ij}^N=\frac{J_j^{\uparrow }-J_{ij}}{J_j^{\uparrow }-J_j^{\downarrow }}$

(4-22)

where in the above equations

$\begin{array}{c}\hfill J_j^{\uparrow }=\max \left\{j_{ij}\right|i=\mathrm{1,2}\dots ,m\}\hfill \\ \hfill J_j^{\downarrow }=\min \left\{j_{ij}\right|i=\mathrm{1,2}\dots ,m\}\hfill \end{array}$

(4-23)

Therefore, the normalized judgment matrix (J^N) would be as follows:

$J^N={\left[J_{ij}^N\right]}_{m\times n}$

(4-24)

4.2.2.4. Calculating the Normalized Weight Matrix

To calculate the normalized weight matrix in the GIA an equation is recommended to determine the pairwise comparison (PC) matrix as follows:

$P{C}_{ij}=1\pm 0.1\delta ,\delta =0,1,2,\dots ,10$

(4-25)

where the plus sign is for the time when the ith criteria is δ degrees more important than the jth criteria, while the minus sign is for the time when the jth criteria is δ degrees more important than the ith criteria.

According to the decision-maker’s opinion and the preceding equation, Table 4.3 is recommended for the linguistic pairwise comparison in the grey-MCDM.

However, the improved GIA proposed by [7] uses a combined weighting method. This method uses an objective weighting method called the entropy information method (EIM) together with a subjective weighting method called the analytic hierarchy process (AHP) to make use of the advantages of both methods. The AHP is mostly based on the knowledge and opinion of the decision-maker while the EIM is based on the technical data and judgment matrix. The combined weighting method blends these two weights together.

The subjective weighting method (AHP) is briefly presented in the following:

$\begin{array}{l}{\rho }_j=\sum _{i=1}^{n}P{C}_{ij}\\ {\rho }_{\max }=\max ({\rho }_1,{\rho }_2,\dots ,{\rho }_n)\\ {\rho }_{\min }=\min ({\rho }_1,{\rho }_2,\dots ,{\rho }_n)\\ \theta ={\rho }_{\max }/{\rho }_{\min }\end{array}$

(4-26)

$b_{ik}=\left\{\begin{array}{ll}1+({\rho }_i-{\rho }_k)/{\rho }_{\min },\hfill & {\rho }_i\ge {\rho }_k,\theta \ne 1\hfill \\ {(1+({\rho }_i-{\rho }_k)/{\rho }_{\min })}^{-1},\hfill & {\rho }_i<{\rho }_k,\theta \ne 1\hfill \\ 0\hfill & \theta =1\hfill \end{array}\right.$

(4-27)

$c_{ik}=\log b_{ik},i,k=1,2,\dots ,n$

(4-28)

$d_{ik}=\frac{1}{n}\sum _{l=1}^n\left(c_{il}-c_{kl}\right)$

(4-29)

${b^{\prime }}_{ik}={10}^{d_{ik}}$

(4-30)

Therefore, the characteristic vector of ${b^{\prime }}_{ik}$ is given as follows:

$E_i=\sqrt[n]{\coprod _{k=1}^n{b^{\prime }}_{ik}},i=1,2,\dots ,n$

(4-31)

Finally, the normalized weight matrix in the AHP method would be as follows:

$\begin{array}{l}{W}_i^1=\frac{E_i}{\sum _{i=1}^n{E}_i}\\ 0\le W_i^1\le 1,\sum _{i=1}^n{W}_i^1=1\end{array}$

(4-32)

The above equation gives the normalized weight of each criterion and subcriterion in the AHP.

To compute the combined weight in the GIA, the EIM is used as the objective weighting method. In the EIM method, the proximity degree (PD) of benefit and cost criteria is calculated as follows:

$P{D}_{ij}=\frac{J_{ij}}{J_j^{\uparrow }},forbenefits$

(4-33)

$P{D}_{ij}=\frac{J_j^{\downarrow }}{J_{ij}},for\cos ts$

(4-34)

Then the PD should be normalized as follows:

$P{D}_{ij}^N=\frac{P{D}_{ij}}{\sum _{j=1}^{m}P{D}_{ij}},\sum _{j=1}^{m}P{D}_{ij}^N=1$

(4-35)

The entropy of information shows the uncertainty of the criteria and can be calculated as follows:

$S_i=-\sum _{j=1}^{m}P{D}_{ij}^N\ln P{D}_{ij}^N$

(4-36)

The entropy should be normalized with the maximum entropy. According to the previous equation the maximum entropy occurs when all the elements of $P{D}_{ij}^N$ become equal; therefore we have

$\begin{array}{l}{S}_{\max }=\ln m\\ S_i^N=S_i/S_{\max }\\ S=\sum _{i=1}^n{S}_i^N\end{array}$

(4-37)

Finally, the weights proposed by the EIM can be calculated:

$\begin{array}{l}{W}_i^2=\frac{1}{n-S}(1-S_i^N)\\ 0\le W_i^2\le 1,\sum _{i=1}^n{W}_i^2=1\end{array}$

(4-38)

Up to this point, the weights of the criteria and subcriteria have been calculated using the AHP and EIM; now a linear combination is used to calculate the combined weight (W^C) used in the GIA:

$W_i^C=\sum _{k=1}^2{\lambda }_k{W_i}^k,\sum _{k=1}^2{\lambda }_k=1$

(4-39)

In the above equation, k = 1 and k = 2 correspond to the weights calculated by the AHP and EIM, respectively. The coefficients λ₁ and λ₂ are calculated as follows:

$({\lambda }_1,{\lambda }_2)=\left(\frac{s_1}{\sum _{k=1}^2{s}_k},\frac{s_2}{\sum _{k=1}^2{s}_k}\right)$

(4-40)

$s_k=\exp \left\{-\left[1+\frac{\mu \sum _{i=1}^n\sum _{j=1}^m{W}_i^k(1-J_{ij}^N)}{1-\mu }\right]\right\},k=1,2$

(4-41)

where μ is the balance coefficient and 0 < μ < 1. The impact of μ on the λ’s and on decision-making will be presented in the case studies at the end this chapter.

4.2.2.5. Finding the Ideal and Anti-ideal Solutions for All Criteria

The case in which all the criteria are the best simultaneously is the ideal solution (I^↑) and the case in which all the criteria are the worst simultaneously would be the anti-ideal solution (I^↓). Therefore, the best and the worst solutions for the GIA are as follows:

$\begin{array}{l}{I}^{\uparrow }=\left[I_1^{\uparrow },I_2^{\uparrow }\dots ,I_n^{\uparrow }\right]\hfill \\ I_j^{\uparrow }=\max \left\{J_{ij}^N\right|i=1,2,\dots ,m\}\hfill \end{array}$

(4-42)

$\begin{array}{l}{I}^{\downarrow }=\left[I_1^{\downarrow },I_2^{\downarrow }\dots ,I_n^{\downarrow }\right]\hfill \\ I_j^{\downarrow }=\min \left\{J_{ij}^N\right|i=1,2,\dots ,m\}\hfill \end{array}$

(4-43)

4.2.2.6. Finding the Weighted Distance from the Ideal and Anti-ideal Solutions

The incidence coefficient between the comparative sequence and the ideal and anti-ideal solutions is calculated as

$d_{ij}^{\uparrow \downarrow }(I_i^{\uparrow \downarrow },J_{ij}^N)=\frac{{\min }_j{\min }_i\left|I_i^{\uparrow \downarrow }-J_{ij}^N\right|+\xi {\max }_j{\max }_i\left|I_i^{\uparrow \downarrow }-J_{ij}^N\right|}{\left|I_i^{\uparrow \downarrow }-J_{ij}^N\right|+\xi {\max }_j{\max }_i\left|I_i^{\uparrow \downarrow }-J_{ij}^N\right|}$

(4-44)

where the min_j min_i $\left|I_i^{\uparrow \downarrow }-J_{ij}^N\right|$ and max_j max_i $\left|I_i^{\uparrow \downarrow }-J_{ij}^N\right|$ are the global minimum and maximum of absolute distance between the normalized judgment criteria from the best and worst solutions. In addition, ξ is the distinguishing coefficient, and 0 < ξ < 1. The impact of the distinguishing coefficient on decision-making is also studied in the case studies at the end of this chapter.

Finally, the weighted distance can be calculated according to the following equation:

$D_i^{\uparrow \downarrow }=\sum _{j=1}^n{\beta }_j{W}_j^C{d}_{ij}^{\uparrow \downarrow },i=\mathrm{1,2},\dots ,m$

(4-45)

where β ∈ {0, 1} is the criteria coefficient and is used for the purpose of single- or multicriteria decision-making. When a criterion is supposed to be counted in the decision-making process the β = 1, otherwise β = 0.

4.2.2.7. Calculating the GIG

The grey incidence grade is used for final decision-making in the GIA and is calculated as follows:

$GI{G}_i=\frac{1}{1+{(D_i^{\downarrow }/D_i^{\uparrow })}^2},i=\mathrm{1,2},\dots ,m$

(4-46)

It is obvious that 0 ≤ GIG_i ≤ 1. Therefore as GIG_i approaches 1, we are approaching the ideal solution; on the contrary if GIG_i approaches zero, we are getting closer to the anti-ideal solution.